So why Prius doesn't handle?

agree 100%
and it is also fun to go through corners on well handling vehicle, car or bike.

front compress is not necessarily a bad thing. It changes steering geometry and if set up properly will help to maintain optimal tire grip through the corners.

There are a couple things more you can do to improve Gen3 handling:
- strut brace (~$115) and perhaps underbrace
- better shocks (performance or even ExelGs)
- camber bolt kit + alignment

You could also add lowering springs to list, but it is not an option for me. As is there is barely enough clearance to clear speed bumps around here.

Out of the list the camber kit and alignment made the biggest difference. It is amazing how much .5-.7deg of negative camber and 1/16-1/8" of toe-out can change car handling. After doing it I am a firm believer that the Gen3 handling deficiencies are mostly due to set up, not due to inherit suspension design/cost cutting. By no means it's a sport car, but it is good enough to handle up to the level of non-performance street rubber.