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 $s = t = 3$,thenthemaximumpossiblenumberofedgesina$C_{2k+1}$-free graph containing no induced copy of $K_{s,t}$ is asymptotically equal to $(t - s + 1)^{1/s}\left(\frac{n}{2}\right)^{2-1/s}$exceptwhen$k = s = t = 2$. This strengthens a result of Allen, Keevash, Sudakov and Verstra"{e}te and answers a question of Loh, Tait and Timmons.