Harry Dresden has described himself as 'easily in the top 40, world-wide, for raw power'. What he lacks is precision, experience, and technique.
His power has never really 'spiked' without outside assistance. Rather, his technique and precision have improved and he has gained experience. He's gotten more powerful as well, but that is because magical ability grows as you exercise it.
He is actually very rarely outmatched. Of the foes he faces in his normal life, he's more than a match for them. The books, however, focus on the times when he's up against serious opposition. Harry throws down with vampires, werewolves, Fae, evil practitioners, etc. These are hardly your run-of-the-mill opponents.
For reference, let's consider a situation in Changes (book 12). I will attempt to keep this spoiler-light, since you likely haven't read it yet.
Harry's fairy Godmother, Lea, is outfitting him. She feels that, for the upcoming event, Harry must look the part. She magically changes his clothing at a whim, and layers the clothes with magical protection. Unlike his coat, this protection covers his whole body (even where the cloth itself doesn't) and can stop rifle rounds with no ill effect.
Harry is miffed - this spell far exceeds the protections he's built into his coat, and Lea has shown little or no effort in doing it. Then he asks how long it will last, remembering that his most recent tweak to his jacket's spell will last almost a year before being refreshed. Lea's spell will last less than 24 hours.
This should be taken as a perfect example of what I started off with: Lea is 2nd in power only to the Winter Queens. She is an immortal who has millennia of experience. Her technique has been refined for centuries or more, and her precision is exceptional. Her spell is outside of anything Harry could accomplish without a lot of time and effort...but he could do it.
To quote his brother, “He's Gandalf on crack and an IV of Red Bull, with a big leather coat and a .44 revolver in his pocket.” Even Gandalf, however, has Sauron and Balrogs. Harry meddles in things that are typically beyond mortal ken, mouths off to literal deities, and spits in the eye of monsters that would give the Faerie Queens pause. When you're going up against things that powerful, you ARE outclassed, even if you happen to be really strong.
Edit: I think I may have thought of the exact moment when the change happened. Spoilers for the end of Dead Beat:
It might, just MIGHT have been when Harry decided he had nothing better to do with his Saturday night than to
Raise a 65-million year old Tyrannosaurus Rex from the dead and ride it through downtown Chicago to save the Wardens, including his former night-terror-inducing boogeyman Morgan, from the Disciples of Kemmler (himself one of the more potent necromancers ever known). He even brought along his very own one-man polka band, to provide background music.
Or maybe it was when he realized that his actions, in addition to saving the Wardens, the city, and possibly the world from a necromantic death god, had compelled Morgan (who spent the first book hounding Harry unjustly, and still caused him the odd nightmare) to have to
wear said polka suit in public.
If you haven't read the books, I'm not joking about a single point. I swear, it makes perfect sense in context.