Growth of products of subsets in finite simple groups

We prove that the product of a subset and a normal subset inside any finite simple non-abelian group $G$ grows rapidly. More precisely, if $A$ and $B$ are two subsets with $B$ normal and neither of them is too large inside $G$, then $\mathrm{\mid }AB\mathrm{\mid }\ge \mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B{\mathrm{\mid }}^{1-\epsilon }$ where $\epsilon >0$ can be taken arbitrarily small. This is a somewhat surprising strengthening of a theorem of Liebeck, Schul, Shalev.