K_(2,t+1)-free graphs with many copies of K_(t,t)

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=\Theta_t(n^2). \] This answers a question of Spiro.