Your plumbing system should look like one of the options on the attached sketch below.
When hooked up to city water, the city water provides all the water flow to your faucets. Your fresh water tank should NOT fill unless the check valve in your water pump is leaking backwards. When not on city water, turn on your water pump and the water pump will pump water from your fresh water tank to all your faucets. You may only be getting a trickle because you have an empty fresh water tank. You should have a gravity freshwater fill on the side of your TT. Should look like this:
Remove the cap and use your hose to fill the tank. Note there's a small round wire mesh on the side at about the 10:30 position; this is the vent line. Don't fill faster than the vent can relieve the pressure or you'll "blow up" your fresh water tank.
If your fresh water tank has enough water in it, another possibility of trickle: you've introduced some plastic shavings from your fresh water tank into the system and it's caught in the faucet strainers. Unscrew them and check for "stuff."