  The Tur\'an function $ex(n,F)$ denotes the maximal number of edges in an
$F$-free graph on $n$ vertices. We consider the function $h_F(n,q)$, the
minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$
edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is
bipartite or colour-critical. In this paper we investigate the simplest
remaining graph $F$, namely, two triangles sharing a vertex, and establish the
asymptotic value of $h_F(n,q)$ for $q=o(n^2)$.
