A student first asked me this question in an American Culture class I taught several years ago. Since then other students have asked similar questions. I sent the question to many American friends, or people I know who live in the U.S., and some of them sent it to other friends of theirs. Some of the responses are below:

Our daughter will be 19 in a few weeks and is in her first year at college.  We think she should make her own decisions, so she has chosen the school she will attend and the courses she will take. We try to be as helpful as possible and while we have always encouraged her to make her own decisions, we have tried to teach her how to make those decisions the best way too.

Probably not all parents feel the same way, and if our daughter was a different kind of person, we might feel differently too.  I think many American children can make good decisions at 18. They will make some mistakes because they are still young, but will learn from their mistakes.
B – Washington State – Western U.S.

It's a gradual thing; at that age, any ties of dependence are based more on economics than law or brute force.
D – Minnesota – Midwest U.S.

I think it's as difficult for American parents as it is for Chinese parents when their children turn 18. It's very hard to stop thinking of them as children, and to allow them to make their own decisions. It's a struggle that lasts a long time in some families. I think this is a universal problem!
E – U.S.A.

Some families allow their children to live at home when they go to the university or begin working. It is expensive to live alone. Other families give their children money to start their lives in their own apartments. Some families have no money to help their children and the children must work very hard.
K – Minnesota – Midwest U.S.

This varies from family to family. Families who have the financial resources tend to help their children into their early 20s. But it's by no means universal, even within a family. My sisters and I all went directly to college; my brother went to work instead, and went back to college years later.

By law, you're independent for most purposes at age 18 (voting, military, most contracts like apartments and college and getting a car loan), and for all purposes at age 21 (alcohol, guardianships, etc.).
F – Boston – New England

For the most part they are nudged out of the nest unless they are going to school ...
C – Asian-Canadian living in Oklahoma, Great Plains

We think it's time for them to go to college and get an education that will allow them to support themselves and contribute something to society or move out and get a job.

Most kids receive ongoing support from their families through college and sometimes beyond because it is increasingly difficult for them to find jobs even with a college education that pay a living wage.
L – Minnesota – Midwest U.S.

Get out of the house and get a job. 
G – U.S.A.

Generally, Americans think that by 18 a child is able to make decisions about how to live their own life.
D – Wisconsin – Midwest U.S.

Some of my friends have had bad times with their families and cut off ties with them when they reach 18. Others stay very close with their families, even when they grow much older.

Many Americans go to college when they are 18, and that tends to make them more independent. At 18, you are legally an adult in the U.S. You are allowed to vote in elections, join the military, get married and a host of other things.
P – Minnesota – Midwest U.S.