I do Crossfit for about two months, during these two months, we have tried as many techniques as possible, like deadlift, snatches, clean and jerk, power clean and thruster.
Ever since I tried thruster and push press, I have pain in the both wrists, usually inside my wrists, it amplifies if I lift weights on my shoulders (so my wrist makes a 90° angle with my forearm) like push press or thruster (max. weight 30kg/66 lbs) also in clean and jerk only in jerk part, although I have no pain while doing deadlifts.
We do always extensive warm-ups, I've already discussed it with my coach, he suggests me to put an empty barbell on my forearm after training and massage it for a while, it actually helped a lot.
I know for a fact that I have very weak wrists, I love weight lifting part so I don't want to stop practicing it.
I'm not quite sure if the pain goes away after a period of time or I need to take action and working on my wrist now. I'm not looking for medical advice here, but rather wrist-strengthening exercise that usually weightlifters do. I've heard that this problem is quite popular among new weightlifters so even if someone can tell me from experience then I'll appreciate it.
I tried wrist supports, they actually helped, but my coach suggested to put them away and let my wrist to have natural movements.
I've checked with my doctor finally (about two weeks after I posted this question here), she told me that I overloaded my wrist muscles and those on my forearm. She suggested of course rest, exercises mentioned bellow in the answers and putting some ice right after workout. And avoid working out the injury. Meanwhile I have to ask my coach to give me exercises which target other muscles in my forearm. I even asked her about carpal tunnel syndrome, as someone suggested here in the comments, she didn't rule out the possibly of I having it, but the chances are very little in my case.
Update 12 October, The combination of wrist band and proper wrist position together with wrist strengthening exercises actually helped a lot. I've got rid of the pain. Although there is still some leftover pain.