I don't think this is an "American" problem.
Almost every day you hear about how parents in Asia are so strict and how wonderful their kids are and how their beating us on PISA and if we don't buckle down and get our kids in line then America will "lose". So we end up with "Tiger Mom" books telling us to watch our kids every minute, to never let them make a mistake, and to criticize the hell out of them should they slip up. Everyone applauds. (OK, not everyone.)
I think there are a lot of reasons we got this way though:
We're fed a steady diet of horror stories about the awful things that can happen to kids if you aren't watching every minute. The internet speculates on how terrible those parents must have been -- they allowed their child to be unsupervised and look what happened!
Your kid makes a nasty post on Facebook? Shoot their computer and post it to Youtube! Get famous as a hero! Appear on the morning talk shows!
Your kid fails a class? Make them hold a sign on the street announcing their crime! Post it to Facebook! Get famous as a hero! Appear on the morning talk shows!
People pat you on the back and tell you you're "doing it right".
If you aren't busy becoming a hero on the internet, you can get busy becoming a criminal:
I saw my neighbor's kids out after dark smoking cigarettes, what do I do? Call CPS!
My neighbor's baby is crying, what do I do? Call CPS!
She allowed her kid to ride the subway alone! Call CPS! She doesn't deserve to have kids! Send her a death threat!
Parents are afraid to let their kid make a mistake because they're afraid they'll get their kid taken away.
There's also the fear that if your kid makes one mistake that their life is essentially over because they'll never get into college and they'll spend their life in a dead end job and it's all your fault.
If you refuse to drug your kid, you're a bad parent.
If your kid tries drugs, you're a bad parent.
I think the biggest problem is that our notion of privacy has changed to the point where a lot of people are fearful of allowing their child any privacy. One forum I look at has a lot of young people on it and they don't seen to understand that privacy is desirable. It's almost like they haven't really done something unless they've announced it publicly.