Some universities in the US have apartments instead of dorms at least. I went to UT Dallas (graduated 2007), where they have apartments. Freshman apartments are 4-bedroom units with each student getting their own bedroom (and pairs of students sharing a bathroom), and apartments available to others are mostly 1-2 bedroom units (with each student getting their own bathroom) with a few 4-bedroom units scattered around.

With that said, I did hear that after I graduated they added dorms for freshmen, but I also heard that the dorms all had single rooms, with the only difference between the dorms and the apartments was that the dorms had communal bathrooms.

For what it's worth, I would not have gone to any school where I would've had to share a bedroom. I'm an intensely private person, and if I couldn't go to college anywhere I wouldn't have to share a bedroom I'd just have entered the workforce at 18 and continued living with my parents till I could afford to move out.