No Brienne is not a better fighter than Jaime Lannister.
Brienne and Jaime fought each other while she was escorting him to King's Landing. Jaime at that point was handcuffed and starved for weeks yet he put up hell of a fight.
But Personally Jaime came to realise she was much stronger than him, while Brienne reached the conclusion that in a fair fight, Nobody stood a chance against the Young Lion.
This is how it went from Jaime's POV:
As the blade slid from the scabbard, he was already pivoting,
bringing the sword around and up in a swift deadly arc. Steel met
steel with a ringing, bone-jarring clang. Somehow Brienne had gotten
her own blade out in time. Jaime laughed. “Very good, wench.”
“Give me the sword, Kingslayer.”
“Oh, I will.” He sprang to his feet and drove at her, the longsword
alive in his hands. Brienne jumped back, parrying, but he followed,
pressing the attack. No sooner did she turn one cut than the next was
upon her. The swords kissed and sprang apart and kissed again.
Jaime’s blood was singing. This was what he was meant for; he never felt so alive as when he was fighting, with death balanced on every
stroke. And with my wrists chained together, the wench may even give
me a contest for a time. His chains forced him to use a two-handed
grip, though of course the weight and reach were less than if the
blade had been a true two-handed greatsword, but what did it matter?
His cousin’s sword was long enough to write an end to this Brienne of
Tarth.
High, low, overhand, he rained down steel upon her. Left, right,
backslash, swinging so hard that sparks flew when the swords came
together, upswing, sideslash, overhand, always attacking, moving into
her, step and slide, strike and step, step and strike, hacking,
slashing, faster, faster, faster...
... until, breathless, he stepped back and let the point of the
sword fall to the ground, giving her a moment of respite. “Not half
bad,” he acknowledged. “For a wench.”
She took a slow deep breath, her eyes watching him warily. “I would
not hurt you, Kingslayer.”
“As if you could.” He whirled the blade back up above his head and
flew at her again, chains rattling.
Jaime could not have said how long he pressed the attack. It might
have been minutes or it might have been hours; time slept when swords
woke. He drove her away from his cousin’s corpse, drove her across the
road, drove her into the trees. She stumbled once on a root she
never saw, and for a moment he thought she was done, but she went to
one knee instead of falling, and never lost a beat. Her sword leapt
up to block a downcut that would have opened her from shoulder to
groin, and then she cut at him, again and again, fighting her way
back to her feet stroke by stroke.
The dance went on. He pinned her against an oak, cursed as she slipped
away, followed her through a shallow brook half-choked with fallen
leaves. Steel rang, steel sang, steel screamed and sparked and
scraped, and the woman started grunting like a sow at every crash,
yet somehow he could not reach her. It was as if she had an iron cage
around her that stopped every blow.
“Not bad at all,” he said when he paused for a second to catch his breath, circling to her right. “For a wench?”
“For a squire, say. A green one.” He laughed a ragged, breathless laugh. “Come on, come on, my sweetling, the music’s still playing.
Might I have this dance, my lady?”
Grunting, she came at him, blade whirling, and suddenly it was Jaime
struggling to keep steel from skin. One of her slashes raked across
his brow, and blood ran down into his right eye. The Others take
her, and Riverrun as well! His skills had gone to rust and rot in
that bloody dungeon, and the chains were no great help either. His
eye closed, his shoulders were going numb from the jarring they’d
taken, and his wrists ached from the weight of chains, manacles, and
sword. His longsword grew heavier with every blow, and Jaime knew
he was not swinging it as quickly as he’d done earlier, nor raising it
as high.
She is stronger than I am. The realization chilled him. Robert had been stronger than him, to be sure. The White Bull Gerold Hightower as
well, in his heyday, and Ser Arthur Dayne. Amongst the living,
Greatjon Umber was stronger, Strongboar of Crakehall most likely, both
Cleganes for a certainty. The Mountain's strength was like nothing
human. It did not matter. With speed and skill, Jaime could beat them
all. But this was a woman. A huge cow of a woman, to be sure, but even
so . . . by rights, she should be the one wearing down.
Instead she forced him back into the brook again, shouting, “Yield!
Throw down the sword!”
A slick stone turned under Jaime’s foot. As he felt himself falling,
he twisted the mischance into a diving lunge. His point scraped past
her parry and bit into her upper thigh. A red flower blossomed, and
Jaime had an instant to savor the sight of her blood before his knee slammed into a rock. The pain was blinding. Brienne splashed
into him and kicked away his sword. “YIELD!”
Jaime drove his shoulder into her legs, bringing her down on top of him. They rolled, kicking and punching until finally she was sitting
astride him. He managed to jerk her dagger from its sheath, but
before he could plunge it into her belly she caught his wrist and
slammed his hands back on a rock so hard he thought she’d wrenched an
arm from its socket. Her other hand spread across his face. “Yield!”
She shoved his head down, held it under, pulled it up. “Yield!” Jaime
spit water into her face. A shove, a splash, and he was under again,
kicking uselessly, fighting to breathe. Up again. “Yield, or I’ll
drown you!”
ASOS - Jaime III
So this is how it happened for Jaime. He began with utmost derision for Brienne and thought he could have beaten her even if he was chained at hands and starved. Brienne's fighting skills made him realise she was a good fighter for a woman. The dance kept moving and he then admitted that she was good enough for a squire. It was just before the end that he realised that she was stronger than him. Although I think he was actually so rattled to be bested by her that he forgot to really consider how much at a disadvantage was he. In an equal fight, as Brienne herself assessed, He would have killed her. Another curious thing to note is how Jaime appears to be in denial by the end, he believed it was not Brienne's skills but rather her strength and that his skills had gone to rust while imprisoned, that was making him lose. Holding off a Swordsman as skilled as Jaime, even at a disadvantage, requires a certain level of skill.
Brienne had (While noting the crucial factors that Jaime was weakened and handcuffed, severely limiting his moves) good reflexes, as Jaime noted happily in the beginning. She brought out her own sword and deflected Jaime's surprise attack just in time. She also had good defensive skills. She skilfully parried each and every attack Jaime launched in the beginning when Jaime was not tired. Nor did she lack for the resolve to keep fighting. Jaime's attack forced her to her knees yet she made her fall a renewed attack. Jaime pinned her against the trees and yet she escaped the deathblow. After that, it was her on the offensive and Jaime on the defensive. This was after all, how she was taught to fight.
Ser Goodwin had taught her to fight cautiously, to conserve her
strength while letting her foes spend theirs in furious attacks. "Men
will always underestimate you," he said, "and their pride will make
them want to vanquish you quickly, lest it be said that a woman tried
them sorely." She had learned the truth of that once she went into the
world. Even Jaime Lannister had come at her that way, in the woods by
Maidenpool. If the gods were good, the Mad Mouse would make the same
mistake. He may be a seasoned knight, she thought, but he is no Jaime Lannister
AFFC: Brienne II
She seems to be holding back a bit as she said she didn't want to hurt Jaime but that couldn't have possibly lasted for long given the ferocity of Jaime's attack. It could be a mixture of both her training and her oath that caused her to remain on the defensive for most part until she saw Jaime was completely exhausted.
Jaime came to give her the respect she deserved by the time they reached King's Landing.
Ser Loras edged around him. "Are you a craven as well as a killer, Brienne? Is that why you ran, with his blood on your hands? Draw your sword, woman!"
"Best hope she doesn't." Jaime blocked his path again. "Or it's like to be your corpse we carry out. The wench is as strong as Gregor Clegane, though not so pretty."
ASOS - Jaime VII
Brienne's own thoughts on the fight went like this:
Brienne remembered her fight with Jaime Lannister in the woods. It had been all that she could do to keep his blade at bay. He was weak from his imprisonment, and chained at the wrists. No knight in the Seven Kingdoms could have stood against him at his full strength, with no chains to hamper him. Jaime had done many wicked things, but the man could fight! His maiming had been monstrously cruel. It was one thing to slay a lion, another to hack his paw off and leave him broken and bewildered.
AFFC - Brienne I
Brienne was better than most knights of Westeros but Jaime Lannister was not most Knights. He was one of those exceptional natural swordsmen who come about in a century or so. Ser Barristan Selmy thought so about Jaime and that says something.
Some of them had been training for the fighting pits when Daenerys
Targaryen took Meereen and freed them from their chains. Those had had
a good acquaintance with sword and spear and battle-axe even before
Ser Barristan got hold of them. A few might well be ready. The boy
from the Basilisk Isles, for a start. Tumco Lho. Black as maester's
ink he was, but fast and strong, the best natural swordsman Selmy had
seen since Jaime Lannister.
ADWD - The Kingbreaker
In the show however, they made it look like as if Brienne was simply holding back to keep her oath and easily defeated Jaime whereas it was a very close thing in the books. Jaime fell when a slick stone got in the way, turned it into a lunge, drew her blood but then was disarmed by Brienne. That didn't stop the fight either, Jaime brought Brienne down with her and they had a fist-to-fist fight and Jaime almost drove her own dagger into her, which is when she finally overpowered him and the Brave Companions caught up with them.
As to how did that specific encounter between Brienne and Loras turned out in the books, this is how it went (The Blue Knight in the excerpt is Brienne):
The blue knight let his broken shield drop to the ground to free his
left arm, and then the Knight of Flowers was on him. The weight of his
steel seemed to hardly diminish the grace and quickness with which Ser
Loras moved, his rainbow cloak swirling about him.
The white horse and the black one wheeled like lovers at a harvest
dance, the riders throwing steel in place of kisses. Longaxe flashed
and morningstar whirled. Both weapons were blunted, yet still they
raised an awful clangor. Shieldless, the blue knight was getting much
the worse of it. Ser Loras rained down blows on his head and
shoulders, to shouts of “Highgarden!” from the throng. The other gave
answer with his morningstar, but whenever the ball came crashing in,
Ser Loras interposed his battered green shield, emblazoned with three
golden roses. When the longaxe caught the blue knight’s hand on the
backswing and sent the morningstar flying from his grasp, the crowd
screamed like a rutting beast. The Knight of Flowers raised his axe
for the final blow.
The blue knight charged into it. The stallions slammed together, the
blunted axehead smashed against the scarred blue breastplate... but
somehow the blue knight had the haft locked between steel-gauntleted
fingers. He wrenched it from Ser Loras’s hand, and suddenly the two
were grappling mount-to-mount, and an instant later they were falling.
As their horses pulled apart, they crashed to the ground with
bone-jarring force. Loras Tyrell, on the bottom, took the brunt of the
impact. The blue knight pulled a long dirk free and flicked open
Tyrell’s visor. The roar of the crowd was too loud for Catelyn to hear
what Ser Loras said, but she saw the word form on his split, bloody
lips. Yield.
ACOK - Catelyn II
Renly did raise Brienne to the Kingsguard when she requested that honour as her prize. Some Tyrell men however claimed that Brienne's pulled a vile trick in pulling Loras down from his horse like that and it was no proper unhorsing.
