A note on induced Tur\'{a}n numbers

Loh, Tait, Timmons and Zhou introduced the notion of induced Tur'{a}n numbers, defining $\mathrm{ex}(n,\{H,F\text{-ind}\})$ to be the greatest number of edges in an $n$-vertex graph with no copy of $H$ and no induced copy of $F$. Their and subsequent work has focussed on $F$ being a complete bipartite graph. In this short note, we complement this focus by asymptotically determining the induced Tur'{a}n number whenever $H$ is not bipartite and $F$ is not an independent set nor a complete bipartite graph.