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