Counting Odd Cycles in Graphs with Bounded Circumference

For an integer $L\ge2$, let $a=\lfloor L/2\rfloor$. Let $H(n,L)$ be the join of $K_a$ and an independent set of order $n-a$, with one extra edge in the independent set when $L$ is odd. We prove that, for every fixed $s\ge3$ and $L\ge2s+2$, and for all sufficiently large $n$, $$\operatorname{ex}(n,C_{2s+1},\mathcal{C}_{\ge L+1}) =N(C_{2s+1},H(n,L)).$$ Together with the recent result of even-cycle by Zhao and Wang~[arXiv:2607.04357, 2026], this settles the conjecture of Zhu, Gy\H{o}ri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)] on counting fixed cycles in graphs with bounded circumference. We also determine the corresponding maximum number of copies of odd cycles when a path is forbidden.