Is side-Apparition not allowed for underage people?

Question

Why bother with flying cars, Floo powder and the Hogwarts Express if parents could just Side-Apparate with their kids anywhere?

Answer 1

Apparition is extremely difficult and many wizards don't do it.

Apparition is one of the riskier forms of magical transportation, more than brooms or Floo powder. Apparating carries the risk of Splinching, and both regular and Side-Along Apparition cause an unpleasant sensation in the wizards being transported this way.

“They left half of themselves behind,’ said Mr Weasley, now spooning large amounts of treacle onto his porridge. ‘So, of course, they were stuck. Couldn’t move either way. Had to wait for the Accidental Magic Reversal Squad to sort them out. Meant a fair old bit of paperwork, I can tell you, what with the Muggles who spotted the body parts they’d left behind …’

Harry had a sudden vision of a pair of legs and an eyeball lying abandoned on the pavement of Privet Drive. ‘Were they OK?’ he asked, startled.

‘Oh yes,’ said Mr Weasley matter-of-factly. ‘But they got a heavy fine, and I don’t think they’ll be trying it again in a hurry. You don’t mess around with Apparition. There are plenty of adult wizards who don’t bother with it. Prefer brooms – slower, but safer.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 6 (The Portkey)}}

Arthur Weasley describes Apparation as something not to be messed with, and says many wizards prefer the safety of brooms instead. Presumably, most wizards would be even more careful when traveling with underage children than they would be alone, so it's likely most wizards wouldn't want to risk Side-Along Apparition with their children or children they care for.

Harry at sixteen reacts badly to Dumbledore transporting him by Apparition.

“Harry felt Dumbledore’s arm twist away from him and redoubled his grip: the next thing he knew, everything went black; he was being pressed very hard from all directions; he could not breathe, there were iron bands tightening around his chest; his eyeballs were being forced back into his head; his eardrums were being pushed deeper into his skull, and then – He gulped great lungfuls of cold night air and opened his streaming eyes. He felt as though he had just been forced through a very tight rubber tube. It was a few seconds before he realised that Privet Drive had vanished. He and Dumbledore were now standing in what appeared to be a deserted village square, in the centre of which stood an old war memorial and a few benches. His comprehension catching up with his senses, Harry realised that he had just Apparated for the first time in his life.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 4 (Horace Slughorn)}}

Dumbledore was a skilled enough wizard where he wouldn't accidentally harm Harry, and Harry is old enough to understand what's happening, but Harry still doesn't feel particularly good after his experience Apparating. Harry says he prefers brooms after that experience.

In addition, Ron gets badly Splinched when Hermione tries to Side-Along Apparate with him.

“Splinched,’ said Hermione, her fingers already busy at Ron’s sleeve, where the blood was wettest and darkest.

Harry watched, horrified, as she tore open Ron’s shirt. He had always thought of Splinching as something comical, but this … his insides crawled unpleasantly as Hermione laid bare Ron’s upper arm, where a great chunk of flesh was missing, scooped cleanly away as though by a knife.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 14 (The Thief)}}

Ron is in pain, missing flesh, and has to have dittany applies to his wounds. In this particular incident, they needed to escape quickly, but in situations that are less dire, it's easy to see why wizards wouldn't risk having this happen.

For the average wizarding family's transportation needs, brooms would likely be sufficient, and come with less risk and discomfort than Apparition.

In addition, Apparating somewhere usually requires being able to clearly visualize that place.

The steps of Apparition typically require the wizard being able to clearly picture the place they're attempting Apparating to. First they have to picture their destination.

“Step one: fix your mind firmly upon the desired destination,’ said Twycross. ‘In this case, the interior of your hoop. Kindly concentrate upon that destination now.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 18 (Birthday Surprises)}}

Then they have to concentrate on trying to occupy the space they're picturing.

“Step two,’ said Twycross, ‘focus your determination to occupy the visualised space! Let your yearning to enter it flood from your mind to every particle of your body!”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 18 (Birthday Surprises)}}

Presumably, then, they might not be able to Apparate somewhere if they can't visualize it adequately. So, for example, wizards who haven't been to the stadium the Quidditch World Cup was held in and can't visualize it wouldn't be able to Apparate there. However, there are instances where wizards do seem to Apparate somewhere they’ve never seen before, like Bellatrix being able to Apparate to Snape’s house though she’s surprised by where it is and how it looks, so it’s not always necessary to have previously visited.

However, it's not illegal to transport an underage wizard this way.

The Ministry recommends it as a way to remove underage wizards from danger, which they wouldn't do if they had made it illegal.

“Review the security arrangements around your house, making sure that all family members are aware of emergency measures such as Shield and Disillusionment Charms and, in the case of under-age family members, Side-Along-Apparition.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 3 (Will and Won’t)}}

In addition, Dumbledore transports Harry this way before he's seventeen.

“You have not, of course, passed your Apparition test?’ he said.

‘No,’ said Harry. ‘I thought you had to be seventeen?’

‘You do,’ said Dumbledore. ‘So you will need to hold on to my arm very tightly. My left, if you don’t mind – as you have noticed, my wand arm is a little fragile at the moment.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 4 (Horace Slughorn)}}

Answer 2

No it's not illegal.

From a Ministry of magic leaflet in the Half Blood Prince:

Review the security arrangements around your house, making sure that all family members are aware of emergency measures such as Shield and disillusionment Charms, and, in the case of underage family members, Side-Along-Apparition
_{Harry Potter and the Half Blood Prince- Will And Won't}

It's probably really difficult.

There are many hints throughout the book that point to apparition being difficult, and I suppose side along apparition is even more complicated.

1. The fact that the ministry fines people for doing it illegally, and you have to pass a test after taking lessons to apparate:

   “You have to pass a test to Apparate?” Harry asked.
   “Oh yes,” said Mr. Weasley, tucking the tickets safely into the back pocket of his jeans. “The Department of Magical Transportation had to fine a couple of people the other day for Apparating without a license. It’s not easy, apparition, and when it’s not done properly it can lead to nasty complications. This pair I’m talking about went and splinched themselves.”
   Everyone around the table except Harry winced.
   “Er — splinched ?” said Harry.
   “They left half of themselves behind,” said Mr. Weasley, ...
   “So, of course, they were stuck. Couldn’t move either way. Had to wait for the Accidental Magic Reversal Squad to sort them out...
   “Were they okay?” he asked, startled.
   “Oh yes,” said Mr. Weasley matter-of-factly. “But they got a heavy fine, and I don’t think they’ll be trying it again in a hurry. You don’t mess around with Apparition. There are plenty of adult wizards who don’t bother with it. Prefer brooms - slower, but safer.”
   _{Harry Potter and the Goblet of Fire- The Portkey}

2. Things can go terribly wrong
   

   There was a horrible screech of pain and everybody looked around, terrified, to see Susan Bones of Hufflepuff wobbling in her hoop with her left leg still standing five feet away where she had started.
   ...
   Three lessons on, Apparition was proving as difficult as ever though a few more people had managed to Splinch themselves.
   _{Harry Potter and the Half Blood Prince- Birthday Surprises}
   
   "...Disarmed the bloke holding mine, and Disapparated. I didn’t do it so well, Splinched myself again" Ron held up his right hand to show two missing fingernails;
   _{Harry Potter and the Deathly Hallows- The Silver Doe}
   
   The moment his eyes fell upon Ron, all other concerns fled Harry’s mind, for blood drenched the whole of Ron’s left side and his face stood out, grayish-white, against the leaf-strewn earth...
   “What’s happened to him?”
   “Splinched,” said Hermione, her fingers already busy at Ron’s sleeve, where the blood was wettest and darkest. Harry watched, horrified, as she tore open Ron’s shirt. He had always thought of Splinching as something comical, but this...His insides crawled unpleasantly as Hermione laid bare Ron’s upper arm, where a great chunk of flesh was missing, scooped cleanly away as though by a knife.
   _{Harry Potter and the Deathly Hallows- The Thief}

Answer 3

I will answer your examples with counter-examples to prove why they need bothering.

Hogwarts Express

Apparition in Hogwarts is forbidden, so parents can only side-appariate their kids to Hogsmeade. After that, boats and thestral-powered carriages are still required. Imagine the crowd there, both children and their parents trying to find a empty carriage or boat.

As you may know, it is usually impossible to Apparate or Disapparate within Hogwarts. The headmaster has lifted this enchantment, purely within the Great Hall, for one hour, so as to enable you to practice. May I emphasize that you will not be able to Apparate outside the walls of this Hall, and that you would be unwise to try.

Harry Potter and The Half Blood Prince,Ch. 18 Birthday Surprises

You still need a reliable transportation method for muggleborns, which they can comprehend by saying " just take your kid to the train station". Muggleborns also do not have anyone to side-appariate to school.

"Er -- I need to be at King's Cross tomorrow to -- to go to Hogwarts."

Harry Potter and The Philosopher's Stone, Ch. 6 The Journey From Platform Nine And Three-quarters

Students also leave their luggages in the train and they are carried separately to school. With parents carrying luggages alongside the children, Hogwarts can not organize such a system.

"Please leave your luggage on the train, it will be taken to the school separately."

Harry Potter and The Philosopher's Stone, Ch. 6 The Journey From Platform Nine And Three-quarters

Floo

You can not Floo in Hogwarts(Umbridge had priviledge).It turns out you can Floo in Hogwarts,which is another reason not to use Apparition. Floo is used mainly to come to the ministry. While an average student does not have to visit the ministry once at all, apparition is still something difficult to do for adults. Other answers explain that part clearly.

Flying Cars

No one actually bothers with flying cars. That is pure Arthur Weasley.

Extra: Brooms

“How’re we getting — wherever we’re going?” Harry asked.
“Brooms,” said Lupin. “Only way. You’re too young to Apparate, they’ll be watching the Floo Network, and it’s more than our life’s worth to set up an unauthorized Portkey.”

---

In the end, I believe Hogwarts Express and Floo is much more reliable than Apparition. Both for the carrier and the carried.