2001 SC1 Saturn Sport Coupe Stalls on 3/4 of a tank left

2001 SC1 Saturn Coupe 145,000 Miles
Stall Problem

Hi, I have looked around here but have not really found an answer to my question. Some seem to have similar problems as mine but not quite the same. Anyways, this problem started about 6 months ago. When ever I have to make a hard stop (example at a yellow light), make a sharp turn or take a tight curve it sometimes stalls. It only does it when its low on gas (usually when there is about a quarter of a tank left.) The first tome this happened I though maybe my fuel gauge was slightly off and I was actually had an empty tank. First off the car started right back up after the stall. After that I drove straight to the gas station. My gas gauge was correct I still had just a little over three gallons in the tank. Since my car still gets an average 28 mpg having 3 gallons left is quite a bit. Anyways I filled it up and I did not have another problem till my tank got down to a quarter again. I really cant figure it out my car runs perfectly fine when its full of gas and even when its low on gas and I stop really slow and smooth it wont die. At one point I let the gas get really low to see how the car would act. The lower the gas got the more trouble I had. Also my gas gauge in my car seems perfectly fine till its get down to just below a quarter of a tank then it goes up and down slightly. I have just been continuing to drive with it like this. Since every time it dies it starts right back up. Its not a problem at all if I keep the tank with at least 4 (it only holds 10-11 gals) gallons of fuel in there.
I would think all this would have something to do with the fuel pump or fuel level. But everyone one I talk too seems to be kind of flabbergasted when I ask them about it. They think like me. If it was the fuel pump why wouldn’t I have a problem all the time? We have a few other ideas but I just thought I would put the question out there.

I would really appreciate any answers or comments any one may have. Sorry its so long just wanted make it all clear. THANKS for your time!