Exact results for generalized extremal problems forbidding an even cycle

We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s\ge 2$ and sufficiently large $n$. Moreover, for $s\in \{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph.