Given graphs H and F with χ(H) < χ(F ), we say that H is weakly F -Turán-good if among n-vertex F -free graphs, a (χ(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 -Turán-good, improving a very recent asymptotic bound due to Grzesik, Győri, Salia and Tompkins. They also showed that for any r there exist graphs that are not weakly K r -Turán-good. We show that for any non-bipartite F there exist graphs that are not weakly F -Turán-good. We also show examples of graphs that are C 2k+1 -Turán-good but not C 2 +1 -Turán-good for every k > .