Much of the side-to-side jerking is a combination of the poor suspension on the trucks (bogies) of the older Gallery train cars and fast running over track switches. The Gallery cars are the ones that look like steel toasters. I imagine there's also quite a bit of hunting / side-to-side movement as well. BART tried to solve this with flat wheels, and after forty years of complaints came to the conclusion that flat wheels wear the track down more and cause significant noise. They've since cut a conical profile (like most rail operators) into the wheels under the guise of reducing noise. I can't hear much difference but the side-to-side motion is quite a bit more dramatic than I remember.
If you take a ride in one of the "newer" Bombardier cars (the octagonal ones) the ride is much smoother and the bumps are almost imperceptible with little swaying. I'd imagine the new trains post-electrification will also be much better on the suspension front.
It's been almost a decade since I've ridden Caltrain with any regularity but I don't remember being particularly enamored with how the Bombardier cars rode. If anything they rode firmer than the Nippon Sharyo cars over the bumps but had less lateral movement as a consequence.
I remember there being quite a bit of variation (there are three distinct batches of them after all) – something you used to see with Muni buses. The German ones (Neoplan) touted their complex (independent) front suspension, and there was a huge variation in ride comfort with some being downright scary. The Hungarian (Ikarus) ones with their simpler solid axle up front seemed more consistently comfortable.