Roads, no. Streets and roads are public because of the impossibility of competition, and because of the inherent civic need.
Highways and freeways in certain cases, yes. Major bridges without question yes. We have private bridges in my area and they work well. The Astoria bridge was so successful that it is now free of toll indefinitely. The state of WA would have been unable to afford the Tacoma Narrows expansion without private investment.
Couple of misconceptions in this thread.
Whether you pay a toll, road tax, or gas tax to fund your roads, it's all mostly the same. There ain't no such thing as a free lunch, i.e. the money has to come from somewhere. You can pay at a toll booth, you can pay at the pump, or you can pay with your annual license renewal. Personally, I like paying at the pump. Partly because that favors fuel-efficient cars but mostly that's because that's the way it's done where I live. It's still a per-mile tax.
Virtually all highways and roads are originally constructed by private contractors and sub-contractors. Maintenance may or may not be done by state and local departments. The statement that all private contractors do shoddy work is ridiculous. And if the state doesn't go after them, in most cases I've seen that because the contract was poorly written (i.e. the state specified the wrong roadbed thickness, materials, etc. -- very common).