If an n-vertex C_{2k+1}-free graph has at least floor((n-r+1)^2/4) + C(r,2) edges, then every odd cycle in it has length at most r, and the bound is sharp.
Brandt, A sufficient condition for all short cycles, Discrete Applied Mathematics 79 (1997), 63–66
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Longest odd cycles in non-bipartite $C_{2k+1}$-free graphs
If an n-vertex C_{2k+1}-free graph has at least floor((n-r+1)^2/4) + C(r,2) edges, then every odd cycle in it has length at most r, and the bound is sharp.