$K_{2,t+1}$-free graphs with many copies of $K_{t,t}$

For every fixed integer $t\ge 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing ${\mathrm{\Omega }}_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that $$\mathrm{e}\mathrm{x}(n,K_{t,t},K_{2,t+1})={\mathrm{\Theta }}_t(n^2)\mathrm{.}$$ This answers a question of Spiro.