I rent a home that has a well, I think it is a submersible but not sure. My problem is that when taking a shower or running water for dishes, the water gets ice cold every time the pump comes on. This has made it just about impossible to use the shower especially now during the winter months and the cold water is extremely cold. I do know the holding tank must be very small, pump will come on three to four times just filling the toilet tank. Please help, my landlord doesn't have a clue.

There should be an air tank. If it comes on that much for a toilet fill, that would be my guess-the tank is waterlogged. Possibly a broken dip tube and the water just short circuits.

It sure sounds to me like there is a faulty check valve... or cold water is somehow crossing over to the hot water side...

Seems to be two problems:

1.
I do know the holding tank must be very small, pump will come on three to four times just filling the toilet tank. That would point to a waterlogged pressure tank. Should be able to put out several gallons at a time before the pump has to cut back on unless the tank is just ridiculously small. This is fairly easy to diagnose if the poster is still listening and interested.

2.
My problem is that when taking a shower or running water for dishes, the water gets ice cold every time the pump comes on. That is strange. From the quote above, it seems that the pump is constantly cutting off/on, so I would think this is not really connected to the pump, unless the poster is saying that when the pump cuts off, the hot water comes back on. Is that the case? If not, then the problem would seem to be with the hot water heater. Broken dip tube, or bad heater element on the bottom.