I can’t answer for America as a whole, but in my circle of friends it is generally accepted as normal. This is true in part because at my school they published the percentage of students who visited the psychological services at least once and it was off the charts—something ridiculously high. Once you have the proof that a lot of your friends have seen a therapist, it kind of becomes normal.
Another thing that has served to normalize therapy for me is that after major disruptive events, my schools (including my high school) have always had therapists on hand for students to talk to. So after 9–11-2001, after a kid at my school committed suicide, after a kid at my college committed suicide, after there were various campus deaths (car accidents, things like that) my schools either brought in therapists for students to talk to or made it clear that they were available. My dean at college even spoke with students one on one about the necessity of mental healthcare. If a student was doing poorly, she’d bring them into the office and try to figure out why. If it was metal health reasons, she’d recommend CAPS (the on campus service) or a few in town psychologists and psychiatrists for if you didn’t want to use CAPS. She was my academic advisor and I found when she checked on how I felt emotionally and socially about the school to be incredibly endearing and supportive.
Mental health is like any other healthcare—if the cost is a concern, there is always a way to get around that. If you attend a college, there is usually some form of free care available to you. If you do not, but live in an area with a therapist degree program, you can still make inexpensive or free appointments, usually.
At the same time as all of this, I have a friend who I believe found it preferable for others to assume she was abusing prescription medications than actually explain she was on an antidepressant. So clearly there are still people who feel it is stigmatized. I’m not sure which way the majority of America trends, but I suspect it falls more towards the stigmatized, but that therapy is becoming less and less stigmatized all the time.