Over the years, i've seen many tv shows that have portrayed children being treated much the same way: they are regarded as helpless and fragile and are often kept away from anything dangerous, no matter how much they want to help in the matter. And that's just how the good guys treat them, the bad guys (or at the very least, complete jerks) frequently see children as helpless pests that they can just push aside and/or force to do whatever they want without any sort of consequence.
I'm certain that you all could come up with reasons why all of this would be reasonable treatment, but the more i think about it, the more i seen less of the POV for this treatment and more of the POV against it, particularly when it's from the child's POV.
D.W. Read from "Arthur" - I can sympathize with how D.W. feels when she is treated like a child. It can't be easy for anyone to hear people tell them that they're too small or too young to do something. It's easy for a child to take that as a sign that people see them as weak and inferior to them.
Gosalyn Mallard from "Darkwing Duck" - Gosalyn wasn't the perfect child, i agree, but it felt like whenever she was trying to do something new, people were trying to get her to do something that went against her basic personality (be the host for a formal gala) and expected her to do so without question or compromise. I think it's safe to say that no one likes being told what to do, especially when they're not allowed to even talk things out.
Jade Chan from "Jackie Chan Adventures" - Again, not the best model of child behavior, i agree, but even still, the way Jade acts is not entirely unreasonable. She knows about the Mobsters, Dragons, Wizards and Shadow creatures that threaten the world around her and yet she's expected to go through life acting like none of it exists. Seriously, how is Jade supposed to go through school life when she knows that beings like Shendu, Daolon Wong and Tarakudo are running amok? Also, the villains basically just treat her like she was nothing, you'd think after what she did to him with the Dragon Talisman, Shendu would want to destroy her too.
Eliza Thornberry from "The Wild Thornberrys" - Eliza's rarely, if ever, allowed to go near any type of animals (even the ones that are known for being docile) and is expected to always be somewhere where her parents can keep an eye on her. This is a problem for two reasons: one; because her parents frequently go out to get near animals of all kinds, so expecting her to not go out and do the same is kind of silly when you think about it. And two; because no child likes to always be in their parents' sight, that might be misconstrued as them being held prisoner.
Zak Saturday from "The Secret Saturdays" - First off, like Eliza, Zak is always kept under constant surveillance and is rarely, if ever, allowed to go anywhere unless someone is watching over him. Again, that must feel like he's in prison. And like Jade, Zak is basically treated like he was nothing by the bad guys, they always go for his parents and treat him as if he wasn't there. Even the other Secret Scientists treat him as if he was no threat. When they came after him, they expected no issues from him alone. None of this can be good for Zak's self-esteem, and the worst part of it is, as soon as Zak's an adult at the most or in his late teens at the very least, he'll be expected to do all sorts of things and then people will start to see him as a threat.