Beka Ergemlidze: Turán 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 $(t − s + 1)^{1/s}(n/2)^{2−1/s}$ except when $k = s = t = 2$.
In this exceptional case we will give some bounds.
This strengthens a result of Allen, Keevash, Sudakov and Verstra ̈ete and answers
a question of Loh, Tait and Timmons.