On Tur\'{a}n problems for Cartesian products of graphs

Let $A,B$ be disjoint sets of sizes $n$ and $m$. Let ${\mathcal Q}$ be a family of quadruples, having $2$ elements from $A$ and $2$ from $B$, such that any subset $S \subseteq A \cup B$ with $|S|=7$, $|S \cap A| \geq 2$ and $|S \cap B| \geq 2$ contains one of the quadruples. We prove that the smallest size of ${\mathcal Q}$ is $(1/16 + O(1/n) + O(1/m)) n^2 m^2$ as $n,m\to\infty$. We also solve asymptotically a more general two-partite Tur\'{a}n problem for quadruples.