  Recently, several hypergraph Tur\'{a}n problems were solved by the powerful
random algebraic method. However, the random algebraic method usually requires
some parameters to be very large, hence we are concerned about how these
Tur\'{a}n numbers depend on such large parameters of the forbidden hypergraphs.
In this paper, we determine the dependence on such specified large constant for
several hypergraph Tur\'{a}n problems. More specifically, for complete
$r$-partite $r$-uniform hypergraphs, we show that if $s_{r}$ is sufficiently
larger than $s_{1},s_{2},\ldots,s_{r-1},$ then
  $$\textup{ex}_{r}(n,K_{s_{1},s_{2},\ldots,s_{r}}^{(r)})=\Theta(s_{r}^{\frac{1}{s_{1}s_{2}\cdots
s_{r-1}}}n^{r-\frac{1}{s_{1}s_{2}\cdots s_{r-1}}}).$$
  For complete bipartite $r$-uniform hypergraphs, we prove that if $s$ is
sufficiently larger than $t,$ we have
  $$\textup{ex}_{r}(n,K_{s,t}^{(r)})=\Theta(s^{\frac{1}{t}}n^{r-\frac{1}{t}}).$$
  In particular, our results imply that the famous
K\H{o}v\'{a}ri--S\'{o}s--Tur\'{a}n's upper bound
$\textup{ex}(n,K_{s,t})=O(t^{\frac{1}{s}}n^{2-\frac{1}{s}})$ has the correct
dependence on large $t$. The main approach is to construct random
multi-hypergraph via a variant of random algebraic method.
