Let $G$ be an infinite group such that the centraliser of any non central element is finite (and bounded).

Is there any structure theorem known about $G$ ?

Such a group seems to be at the other extreme of an FC-group (whose centralisers all have finite index). I can add the following requirements alltogether if need be : 1- $G$ has finitely many conjugacy classes. 2- $G$ has a trivial centre. 3- $G$ has no involution. 4- $G$ is simple.