Thicker sway bars are basically like stiffer springs, but they transfer force from one side of the car to the other. In a turn the it makes the force of the car leaning on to the outside tire and the the force of the inside tire wanting to lift work against each other to sort of cancel out body roll.

The job of the shocks is to keep the suspenion from bouncing around too much with all the inputs from spring(s)--including sway bars. I mean sway bars aren't going to kill your stock shocks or something but you generally want shocks to be the first thing you upgrade since they have the most effect on how the car handles.