Let's put it another way.
As the force of the turn (I don't care if it's centrifugal or centripedal, although I do know that Hack is correct) causes the body of the car to lean, think about what happens:
On the outside of the turn, the body tries to move DOWN (closer to the tires). On the inside of the turn, the body tries to move UP (away from the tires). It doesn't matter if the suspension is independent or solid-axle; it's the springs that are compressing on the outside wheels that allows this to happen. The wheels stay "flat" (because they stay on the ground, unless you're driving like Hack, but that's another story....), while the body leans.
Swaybars try to keep the body aligned with the ground and the wheels. Swaybars are a giant, U-shaped spring. One end of the U attaches at an inside wheel, the other end attaches at the outside wheel. The middle of the U is attached to the body. Imagine that the "U" is lying flat, parallel to the ground, with the arms pointing forward and the curve pointing toward the trunk.
As one side of the car tries to move down toward the wheel, that arm of the U-spring tries to resist, pushing the body back up. At the same time, the other side of the body is trying to move up, but that arm of the U-spring is pushing down, trying to keep the body down. The two arms always try to push in opposite directions.
In the middle, where the U connects to the body, the mounting points allow the whole contraption to swivel. That way, the swaybar has minimal effect on the way that the suspension handles bumps. Hit a speedbump, and both wheels go up (closer to the body), and the swaybar simply pivots on its center connection. The regular suspension springs do all the work soaking up the jolt.
On rough and uneven surfaces,where the tires are trying to bounce in all sorts of directions with the independent suspension, the swaybar has the effect of "semi-tying" the wheels together. To put it another way, the swaybar tries to limit the amount of independence the wheels have. So a somwhat harsher ride can result, under those conditions.