The check valve in the water pump is leaking.
1) Are you using a pressure regulator? High campground water pressure can damage the pump and other easy targets like the toilet flush valve and faucets.
2) Turn off and remove the city water hose. Open a faucet until the water stops running; then close it.
3) Run the pump to pressurize the system. Does the pump keep running or short cycle with everything closed? (Short Cycle - keep turning on and off - not hold pressure)
4) If the water system won't stay pressurized, the check valve has failed. You can rebuild the pump you have with a pump kit or replace it.
Pump Repair Kits - SHURflo
Lou and Laura with Bella - German Short Hair Pointer
2008 GMC Sierra 2500HD Crewcab SB Allison Duramax
2010 Flagstaff 8526RLWS - Superglide 3300
HAM CALLSIGN - KC3FFW