The private universities are rarely better than the public ones outside of a couple of narrow fields. Certainly if your want to study a STEM subject, Economics, Medicine or most humanities the public universities are at least as good and probably better than the private ones. The only area where private universities arguably have an edge is in Business and Management subjects.

And even the most expensive private universities are 'only' $10-15k a year, which while certainly a lot of money, is quite cheap compared to the US. Combine this with a relatively high minimum wage and it's quite possible to pay the tuition with a part time job while studying.