I don't know that cars in the US necessarily have softer suspension than their European counterparts, but I can attest to the fact that it is not due to the rural population. Much of the rural population will be driving pickup trucks or truck-based SUVs, which have much stiffer suspension than a normal car.
I know that older Cadillac models would have very soft suspension (that's why people refer to them as boats, they take bumps like they were riding waves). I can't attest to that being a factor in any higher incidence of rollovers. If there was in fact a higher percentage of rollover accidents in the US, I would put more money on this being due to the fact that the US really likes truck-based SUVs (Jeep Grand Cherokee, Jeep Wrangler, Chevy Blazer/Tahoe, Toyota 4Runner, etc) with lots of weight really high up. My 1998 4Runner even had a warning sticker saying you can't corner like you would in a car without risk of rollover.
In short, people buy cars they don't know how to drive.