  We obtain some new upper bounds on the maximum number $f(n)$ of edges in
$n$-vertex graphs without containing cycles of length four. This leads to an
asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as
well as a disproof of a conjecture of Erd\H{o}s from 1970s which asserts that
$f(n)=\frac12 n^{3/2}+\frac14 n+o(n)$.
