  The generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of
copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where
$\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when
the extremal graph is a complete $(\chi(F)-1)$-partite graph. We obtain
multiple exact results with other kinds of extremal graphs.
