The main fuel pump does need to work a bit harder as the tank level drops. That's the primary purpose of the in-tank pre-pump. If the pre-pump has failed or if the main pump is worn or if there is any restriction in normal fuel flow (like the main fuel filter, in-tank filter sock, fuel pressure regulator or return line) then that could explain things.

The gauge needle fluctuating as you go around a corner may well be a separate issue -although in physical proximity the two systems aren't connected. It could be that the in-tank fuel gauge sending unit is worn and the electrical contacts on the float are somehow moving away from the resistor coil when you do a hard turn. However, from my own experience and what little has ever been posted that isn't a common problem. What is likely is an electrical problem. A common fuel gauge problem for your car is a loose connection at the instrument cluster (check the FAQ and/or archives). You could even have an intermittent short at a pinched wire (going under the rear seat or where the cable goes through the chassis before the tank). However, something that is occurring to me is a faulty (loose) ground at the rear of the car. If that ground point is shared by the pre-pump then that might explain all your problems.
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Dave -not to be confused with a real expert, just goofing around at this