Simonovits's supersaturation conjecture for bipartite graphs

Let $F$ be a bipartite graph such that

$$\operatorname{ex}(n,F)=O(n^{2-\alpha})$$

for some $0<\alpha<1$. Let $G$ be an $n$-vertex graph, and write $v(F)$ and $e(F)$ for the numbers of vertices and edges of $F$. Simonovits's supersaturation conjecture. There exist positive constants $c$ and $c'$ such that if $e(G)\geq cn^{2-\alpha}$, then $G$ contains at least

$$c'\frac{e(G)^{e(F)}}{n^{2e(F)-v(F)}}$$

copies of $F$.

The bound has the order of magnitude predicted by a random graph with $e(G)$ edges. Simonovits proved the conjecture for even cycles, and Erdős and Simonovits proved it for paths and several other graphs, but the general bipartite case remains open.