A subquadratic bound for generalized Turán numbers of odd cycles

For a graph $H$ and a family of graphs $\mathcal{F}$, let $\text{ex}(n,H,\mathcal{F})$ denote the maximum number of copies of $H$ in an $\mathcal{F}$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set ${\mathcal{C}}_r=\{C_3,C_4,\dots ,C_r\},$ and set ${\mathcal{C}}_2=\mathrm{\varnothing }$. In this paper, we prove that, for all integers $l>k\ge 2$, $$\text{ex}(n,C_{2k+1},{\mathcal{C}}_{2k}\cup \{C_{2l+1}\})=O_{k,l}\left(n^{2-\frac{1}{k(k+1)(l-k)}}\right)\mathrm{.}$$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Győri, Methuku, and Vizer.