  We present a new proof of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem that
$ex(n, K_{s,t}) = O(n^{2-1/t})$ for $s, t \geq 2$. The new proof is elementary,
avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let
$ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform
hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph
obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing
every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e
\cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph
on $s$ vertices with $s$ edges, then $K_{H, t} = K_{s, t}$.
  We prove that $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t} + t n^d)$
for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n,
H) = o(n^d)$. Thus $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ for any $d$-uniform
hypergraph $H$ with at least two edges such that $ex_d(n, H) = O(n^{d-1})$,
which implies the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem in the $d = 1$ case.
This also implies that $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ when $H$ is a
$d$-uniform hypergraph with at least two edges in which all edges are pairwise
disjoint, which generalizes an upper bound proved by Mubayi and Verstra\"{e}te
(JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Tur\'{a}n
problems.
