Let $r \leq s$ be fixed natural numbers. Then by the Kővári–Sós–Turán theorem, any graph on $n$ vertices with at least $cn^{2-\frac{1}{r}}$ edges contains a complete bipartite subgraph $K\_{r,s}$ for a constant $c.$


I was wondering if we can say that if we have at least $cn^{2-\frac{1}{r}}+1$ edges, then there are at least $c'n$ complete bipartite subgraphs $K\_{r,s}$, for a constant $c'$?


For example, if a graph has at least $\frac{n^2}{4}+1$ edges, then it has at least $n$ triangles!
I'm pretty sure the answer to my question should be Yes, but I do not know how to prove it.



Kovari-Sos-Turan theorem
The appropriate search term is "supersaturation". I think Theorem 11.1 in 


Furedi, Z., Simonovits, M. (2013). The history of degenerate (bipartite) extremal graph problems. In Erdos Centennial (pp. 169-264). [arxiv:1306.5167](https://arxiv.org/abs/1306.5167)


implies the existence of a constants $c\_1=c\_1(r,s)$ and $c\_2=c\_2(r,s)$ such that a graph with $m>c\_2n^{2-1/r}$ edges contains at least
$c\_1c\_2^{rs}n^r$
copies of $K\_{r,s}$. 


