They could turn it over to private interprise but would they have offices in all those small towns. FedEx and UPS wouldn't do to well either if they had to have an office everywhere there is a Post Office.
Norway har partially privatized postal services. The post offices are gone, and what they did is now handled by local grocery stores. Service has improved and small places that wouldnt have a chance to keep an old style post office has service.
Also banks are doing the same (for the very few things one need a bankoffice for).
What I find strange is that US is still using checks, money orders and sending bills by mail, time to go digital!