  Given graphs $H$ and $F$ with $\chi(H)<\chi(F)$, we say that $H$ is weakly
$F$-Tur\'an-good if among $n$-vertex $F$-free graphs, a $(\chi(F)-1)$-partite
graph contains the most copies of $H$. Let $H$ be a bipartite graph that
contains a complete bipartite subgraph $K$ such that each vertex of $H$ is
adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Tur\'an-good,
improving a very recent asymptotic bound due to Grzesik, Gy\H ori, Salia and
Tompkins. They also showed that for any $r$ there exist graphs that are not
weakly $K_r$-Tur\'an-good. We show that for any non-bipartite $F$ there exists
graphs that are not weakly $F$-Tur\'an-good. We also show examples of graphs
that are $C_{2k+1}$-Tur\'an-good but not $C_{2\ell+1}$-Tur\'an-good for every
$k>\ell$.
