My understanding in the simplest terms is the following:
In a right turn, weight transfer results in the left side of the cars suspension to dive and the right side to rise. The sway bar works to equalize the reaction from one side to the other. If one side moves up or down, the other side follows. This is why, in most cases, a sway bar usually doesn't make a cars ride degrade in the way that larger torsion bars can.