2003 Toyota Avalon - Kicks to the side over bumps

Hello -
I am so frustrated and hoping (as everyone who posts here) that you can help diagnose my issue. My vehicle has a distinct side to side kick when it hits bumps/pot holes or goes over a bump with the brakes applied; like when approaching a red light. It feels almost like the leg action of a bowler at the end of their follow through. I have had all the tires replaced, and sway bar checked, and as much as I trust my small town mechanic I am not getting a clear answer. The car has such a horrible ride and I can take driving it anymore. Please help me! Thank you so much.

Just guessing here, but might be time for new struts.

Might also be time for new springs and jounce bumpers as well. A set of quick struts which come assembled with all new struts, springs mounts and bumpers might be the best solution.