  In 1991 Gy\H ori, Pach, and Simonovits proved that for any bipartite graph
$H$ containing a matching avoiding at most 1 vertex, the maximum number of
copies of $H$ in any large enough triangle-free graph is achieved in a balanced
complete bipartite graph. In this paper we improve their result by showing that
if $H$ is a bipartite graph containing a matching of size $x$ and at most
$\frac{1}{2}\sqrt{x-1}$ unmatched vertices, then the maximum number of copies
of $H$ in any large enough triangle-free graph is achieved in a complete
bipartite graph. We also prove that such a statement cannot hold if the number
of unmatched vertices is $\Omega(x)$.
