I live on a very long (two miles) private road that I jointly build and maintain with an association, etc. There is no government role whatsoever, and we even have bridges.
So it would appear that my experience validates your libertarian fantasy world. Look - we're actually doing it!
I've got sour news for you. Even a wealthy, well educated collection of fair minded and well intentioned people find it very difficult to maintain a dirt/gravel road to an acceptable condition. It's very difficult and costly. We are right on the edge of success or failure in this maintenance.
Further, if we were to actually pave the road (a true fantasy) it would be completely, totally unthinkable with regard to cost. Asphalt is paid for by the square foot and it isn't cheap. When you drive by a very quick section of asphalt road being resurfaced ... say 200-400 meters or something - very short ... understand that that is millions of dollars in asphalt work. Millions.
Can you build some roads and bridges without the government ? Sure, of course you can. Will they approach, in any category, even the bare minimum acceptable standard that you hold ? No.