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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 $d$-uniformhypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraphobtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacingevery 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 hypergraphon $s$ 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 $d$-uniformhypergraph $H$ 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 pairwisedisjoint, 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}nproblems.

A generalization of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem