I noticed that when I use whammy bar and press it harder, my guitar's bridge rise up from guitar's body. Is that normal?
yes that is normal.
2002 PRS CE22
2013 G&L ASAT Deluxe
2009 Epiphone G-400 (SH-4)
Marshall JCM2000 DSL100
Krank 1980 Jr 20watt
Krank Rev 4x12 (eminence V12)
GFS Greenie/Digitech Bad Monkey
Morley Bad Horsie 2
MXR Smart Gate
Depends on the tremolo.

Guitars based on a knife-edge trem (like a Floyd) or on having the front of the trem nailed down to the guitar and the rear end floating will react like that. Those guitars incorporate the bridge into the trem.

A Kahler will not (the action will stay where it is no matter how hard you wail on it), nor will a Bigsby. Same is true for the wide number of trems that are not bridges themselves, but that operate the trem from behind the actual bridge.