Explicit thresholds in a generalized Turán problem for $K_{3,t}$-free graphs

For graphs $F$ and $H$, let $\text{ex}(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Janzer, Longbrake and Yepremyan recently proved that, for fixed $3<a\le b$ and sufficiently large $t$, $$\text{ex}(n,K_{a,b},K_{3,t})=\mathrm{\Theta }(n^3)\mathrm{.}$$ We make their threshold explicit, showing that this conclusion holds for all $t\ge \tau (b):=2\max \{3,\lceil b\mathrm{/}2\rceil \}+1.$ In particular, for every even $b\ge 6$, this matches the necessary threshold $t=b+1$. The main new ingredient is an explicit finite-field point set whose plane sections are controlled directly, rather than through a general bounded-complexity algebraic lemma. This direct line-and-conic section analysis gives the required $K_{3,t}$-freeness while preserving many coplanar $b$-element subsets.