  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.
