Here's the problem. Temp needle started climbing to the top while engine was working. When I turn the engine off and I just give key contact, temp gauge is as usual, slightly above middle. The top hose going from radiator to the engine is very hot, yet the cooler (specially on the bottom) is hardly warm.
I've opened the radiator cap and some water and steam came out. After it cooled down again, I've put water to fill the radiator.
The fan engages, but it seems there's no water circualtion. I've changed my timing belt and water pump 4k miles ago (6.5k km). Could it be the thermostat or could it be faulty water pump? Any advice is appreciated.

open the drain plug on the radiator see if the radiator is clogged. you said the bottoms barely warm while the tops hot sounds like the radiator probably clogged.

Left the car at office, will see that tomorrow. Would clogged radiator spit hot water and steam when I opened it's cap?
Here's the pic of radiator heat, maybe it gives you a clue if it is clogged indeed.

I wasn't sure would there be any circulation if it was clogged... It was a faulty thermostat, replaced now, temps are ok.

Posting in brz thread.