1) If it's expensive to do for the private sector, it's expensive to do for the government, no part of the cost equation changes.
2) It only seems "cheaper" b/c the private sector would have to ask for money by providing enough value enough for you to part with your money, while government doesn't, and raises money by pointing guns at people. However, to avoid the free-rider problem, you can use assurance contracts to raise the construction money and tolls to pay for maintenance, similar to what governments would do anyways.
3) American infrastructure was rated a D+ by the ASCE, so in what way is having government the sole provider of infrastructure a guarantee of quality?
4) slightly unrelated, but do you really think we'd have the same level of urban sprawl, needing a car to get anywhere in suburbia and such a heavy reliance on fossil fuels if roads were built through market forces rather than government subsidy?