HI, all.
There's two things happening here.
1. City water is filling your fresh water tank, which - in turn - overflows onto the ground.
2. When on tank/pump, pump is unable to provide adequate pressure ("dismal," as Holly mentioned).
It's likely that these are related, but they seem like two separate issues. Seems like both need to be addressed.
#1 is a common problem, we see it often - this is what most people are suggesting. I've been getting a lot of mileage out of this diagram lately. It's got a couple of features that your system doesn't have, but it will work to show the theory. Check it out and see if this doesn't help explain things:
Look at the green line. Water comes IN at your city water connection (on the left), flows to the right and down/counterclockwise down to where it T's into the supply line... and then continues out to the rest of the trailer (in this case, through an inline filter, out to the all the cold water supplies and to the hot water heater....
When you're on city water, your entire plumbing system is pressurized by the source line and you get water flowing to your outlets when you open them. That pressure that pushes all that water OUT the green line ALSO puts pressure onto the orange side of line, pushing BACKwards against the pump (in this case, pushing to the left/clockwise).
This is anticipated and by design. Normally, a one-way check valve AT the water pump stops city water from flowing backwards through the pump, and then UPstream (the wrong way) into your freshwater tank, and then - as you discovered - out the overflow vent and/or the sidewall gravity port.
This one way check valve sometimes gets stuck open. Sometimes it's because of sediment, sometimes some buildup or crud, sometimes it just gets stuck and/or worn out. When the valve gets stuck open, water from the city line (pressurized side) is pushed backwards through the pump, slowing making its way to (and through) the fresh tank.
The fix for this can be pretty simple. You can send water back and forth through the checkvalve to exercise it, or unstick it. To do this, simply move back and forth from pump to city water, and let some water flow from a faucet using both methods.
I don't mean to belabor the point, here's some specifics:
Let's say you're starting on city water (pump should be OFF, as it should always be when you're on city water).
1. Open any faucet (cold water will do) and let a little water run. Close faucet.
2. Shut OFF city water valve. Turn ON water pump.
3. Open a faucet and let a little water run. Close faucet.
4. Shut OFF water pump. Turn ON city water valve.
5. Repeat.
Doing this a few times may cause the valve to unstick, freeing any crud or sediment in it, or causing it to reseat itself.
If this doesn't work, you can look to replace the one-way valve, or - in lieue of that - work around it, and install a definite cut off valve. This would serve the SAME purpose as the one-way check valve, but would be more positive.
You can put it on either side of the pump (likely right up against it), but I'm sorta with 5Picker on this, because there are instances where
I will use either my pump OR city water. This aligns with the cardinal rule of campers - if it works for YOU and YOUR use case, then do THAT. What works for you might not work for someone else.
If you are ALWAYS on city water and NEVER on pump (looking at you, Jarufus), then I might be convinced to put the shutoff valve on the outlet side, which would separate the pump entirely from the fresh side, elminating the pump (and all its connections and gaskets and whatnot) as sources of leaks and problems. It'd be installed on the orange line to the RIGHT of the pump in the diagram shown (ideally right up next to it, between the pump and *any* T for any other source/path for water).
For ME and my use case - if I was going to install one - I'd put it on the INTAKE side of the pump (for the system benefit 5picker mentioned: if you forget to open it for when you are on your tank, the pump will continue to run letting you know something was not okay). So in my case it'd be installed on the orange line to the LEFT of the pump (again, ideally right up next to it, between the pump and any T where the (optional) antifreeze intake is shown). That valve will absolutely keep city water from flowing back into your fresh tank (and out onto the ground).
NOW, second issue is that your pump doesn't seem to be providing adequate pressure. I don't have a real grasp on that one yet, and MIGHT be helped with some attention to the system, so I suggest we address issue #1, see how that does, and then try to tackle #2.
Just my .02. I could be - and, as many of you have appropriately pointed out - often am wrong.