Well pump pressure switch cuts on and off when I turn water on. I thought pump turned on at approx. 20psi and run until water was turned off.

A pressure switch turn on pump when pressure drops to a set pressure, then turn off when pressure increases about 20 pounds. It is adjustable.

Other things can affect the operation of the system.

Have an assistant open and then close faucet while you watch pressure gauge. Tell us what pressure gauge reads before opening faucet, what it reads with faucet is open, how long it takes to get back to the original pressure.

Also tell what size tank you have.