Can someone explain to me why these planes just don't turn about their center of gravity?
The thrust force from the engine should generate some clockwise moment, right?


If we take your photo as a reference (and thus we intend "clockwise in the picture plane"), yes, it is. It is called pitch moment, because it would change the pitch of the aircraft (in your case, a A-90_Orlyonok), and in the case shown in your picture is negative, because it would make the aircraft pitch down.


  Why wont it turn and just crash then? I guess the wings generate some sort of anti clock wise moment? 


Because the wings and the horizontal stabilizer produce a total net pitch moment that counteracts the one generated by the engine.

They do so by having the center of lift in a different location w.r.t. the center of gravity.



The "normal" configuration for airliners would have an opposite pitch moment due to the engines, since in that case the engines are usually below the wings (and, being usually low wings, below the center of gravity). It is well illustrated in this image taken from the FAA: