  For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the
maximum number of edges in an $n$-vertex graph which contains none of the
members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal
graph theory asks to determine the function $ex(n,\{C_3,C_4\})$. Here we give a
new construction for dense graphs of girth at least five with arbitrary number
of vertices, providing the first improvement on the lower bound of
$ex(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer
to a problem in Chung-Graham [3].
