prove that every graph on $n$ vertices, $n\geq 4$, with more than $2n^{3/2}$ edges has girth of at most 4

Question

prove that every graph on $n$ vertices, $n\geq 4$, with more than $2n^{3/2}$ edges has girth of at most 4

Given the fact that every simple graph that satisfies $\sum_{v\in V} {{d(v)}\choose{2}} > (m-1){{n}\choose 2}$ contains $K_{2,m}$ how can you prove this? I mean for $n<18$ it follows that ${{n}\choose 2} < 2n^{3/2}$ so it seems weird it would even have this many edges.

Thanks.