Huh, this is very interesting...I used to think about stuff like this a lot, during my YA novel-reading days. ^^
I really think this depends on the power/magic in question. If the magnitude of power that someone can have is completely random (like, one person is born with the power of invisibility, and another is born with the power to generate BLACK HOLES) the only way to keep control would be to either (a) find out how the powers work and have an option to dial them down/confiscate them if necessary, (b) develop a screening process for those who could be a "danger to society" and start brainwashing them from a young age, or (c) eliminate those who have powers indiscriminately.
If none of those are possible, society will inevitably be forced to follow the whims of whoever has the most power and the most willingness to use it. =/
Now if the power has a sort of relatively-harmless baseline from which people can grow (like bending in the Avatar universe) then you can be a little more lax...after all, the odds that someone will be skilled enough to use their power in an especially harmful way are rather low; you can deal with those instances as they occur (like Aang taking Yakone's bending away).
Institutions for training people to be skilled enough to cause serious harm will probably arise at some point, but as long as those are monitored and restricted, it shouldn't be too much of a problem. I mean, there are plenty of people in the world now who can hack and build bombs and 3D-print guns, but the world hasn't fallen into chaos yet. The social contract is stronger than you think~
...Yeah, I wouldn't count on that. ^^ Somewhere, someone will have a problem with being told what to do about ANYTHING, it happens all the time.
But like I said, the social contract is strong: believe it or not, 90% of the people in this world would rather just live their lives peacefully according to the rules than cause trouble, no matter what those rules are. There are plenty of situations (often pre-revolutionary...) where people could and probably should defy the rules for the common good, but choose to obey them because obeying simply isn't as risky as taking a stand.
Regardless, all of the control strategies I mentioned above have their faults, ways they could easily be exploited by people with bad intentions, and/or met with disapproval by the public. But that's what makes writing fiction interesting, right? ^^ Diving into those awful spaces and seeing where they lead; imagining what could happen in a worst-case scenario.
I think it's also relevant to add that a lot of these things are simply ignored in more cartoony, humorous spaces...you know, the kinds of stories where if there's an explosion no one is burned alive or has their limbs blown off; they just get covered in soot and cough? ^^ A world with unrealistically light consequences will rarely give rise to true strife, no matter how many teenagers are running around with superhuman powers.