  For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let
$\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of
$\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination
of Tur\'{a}n number $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ has become the
fundamental core problem in extremal graph theory ever since the pioneering
work Tur\'{a}n's Theorem was published in $1941$. Although we have some rich
results for the simple graph case, only sporadic results have been known for
the hypergraph Tur\'{a}n problems. In this paper, we mainly focus on the
function $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ when $\mathcal{H}$ is one
of two different hypergraph extensions of the complete bipartite graph
$K_{s,t}$. The first extension is the complete bipartite $r$-graph
$K_{s,t}^{(r)}$, which was introduced by Mubayi and Verstra\"{e}te~[J. Combin.
Theory Ser. A, 106: 237--253, 2004]. Using the powerful random algebraic
method, we show that if $s$ is sufficiently larger than $t$, then
\[\text{ex}_{r}(n,\mathcal{T},K_{s,t}^{(r)})=\Omega(n^{v-\frac{e}{t}}),\] where
$\mathcal{T}$ is an $r$-graph with $v$ vertices and $e$ edges. In particular,
when $\mathcal{T}$ is an edge or some specified complete bipartite $r$-graph,
we can determine their asymptotics.
  The second important extension is the complete $r$-partite $r$-graph
$K_{s_{1},s_{2},\ldots,s_{r}}^{(r)}$, which has been widely studied. When
$r=3$, we provide an explicit construction giving
\[\text{ex}_{3}(n,K_{2,2,7}^{(3)})\geqslant\frac{1}{27}n^{\frac{19}{7}}+o(n^{\frac{19}{7}}).\]
Our construction is based on the Norm graph, and improves the lower bound
$\Omega(n^{\frac{73}{27}})$ obtained by probabilistic method.
