Forbidding $K_{2,t}$ Traces in Triple Systems

Let $H$ and $F$ be hypergraphs. We say {\em contains as a trace} if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ subhypergraph isomorphic to $F$. In this paper we give an upper bound on the number of edges in $3$-uniform hypergraph does not contain $K_{2,t}$ trace when $t$ is large. particular, show that
 $$\lim_{t\to \infty}\lim_{n\to \infty} \frac{\mathrm{ex}(n, \mathrm{Tr}_3(K_{2,t}))}{t^{3/2}n^{3/2}} = \frac{1}{6}.$$
 Moreover, $\frac{1}{2} n^{3/2} + o(n^{3/2}) \leqslant \mathrm{ex}(n, \mathrm{Tr}_3(C_4)) \frac{5}{6} o(n^{3/2})$.