Can driving over these humps on the road damage car(pictures included)?

It shouldn't. You shocks and struts might be the issue if your car is bouncing.

Yeah go over slower if you have to.

Hi
if you are bouncing over speed bumps a slow down and B get some new shock absorbers as yours have given up on you so you would easily get into an accident with poor suspension.

Yes it can. Crawl under your car and take a look.

will depend on hump shape .height .ramp up and back . weight of car etc.
speed is a judgemental thing. knowing car an dhow it behaves going over them etc plus judging hump itself.
rule is ' if in doubt- go slower'.
many cars suffer from heavy impacts on suspension etc. damage can occur. bouncing though? ought not to occur unless we have weak dampers?
may be time to have them checked out.