Tur\'an number of an induced complete bipartite graph plus an odd cycle

Let $k \ge 2$ be an integer. We show that if $s = 2$ and $t \ge 2$, or $s = t = 3$, then the maximum possible number of edges in a $C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s + 1)^{1/s}\left(\frac{n}{2}\right)^{2-1/s}$ except when $k = s = t = 2$. This strengthens a result of Allen, Keevash, Sudakov and Verstra\"{e}te and answers a question of Loh, Tait and Timmons.