Asymptotics for the Tur\'an number of Berge-$K_{2,t}$

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Tur\'an number of Berge-$\mathcal{F}$ is the maximum possible number of edges in an $r$-uniform hypergraph on $n$ vertices containing no Berge-$F$ as a subhypergraph (for every $F \in \mathcal{F}$) and is denoted by $ex_r(n,\mathcal{F})$. We determine the asymptotics for the Tur\'an number of Berge-$K_{2,t}$ by showing $$ex_3(n,K_{2,t})=\frac{1}{6}(t-1)^{3/2} \cdot n^{3/2}(1+o(1))$$ for any given $t \ge 7$. We study the analogous question for linear hypergraphs and show that $$ex_3(n,\{C_2, K_{2,t}\}) = \frac{1}{6}\sqrt{t-1} \cdot n^{3/2}(1+o_{t}(1)).$$ We also prove general upper and lower bounds on the Tur\'an numbers of a class of graphs including $ex_r(n, K_{2,t})$, $ex_r(n,\{C_2, K_{2,t}\})$, and $ex_r(n, C_{2k})$ for $r \ge 3$. Our bounds improve results of Gerbner and Palmer, F\"uredi and \"Ozkahya, Timmons, and provide a new proof of a result of Jiang and Ma.