  Let $\mathbb{P}^s$ be the $s$-dimensional complex projective space, and let
$X, Y$ be two non-empty open subsets of $\mathbb{P}^s$ in the Zariski topology.
A hypersurface $H$ in $\mathbb{P}^s\times\mathbb{P}^s$ induces a bipartite
graph $G$ as follows: the partite sets of $G$ are $X$ and $Y$, and the edge set
is defined by $\overline{u}\sim\overline{v}$ if and only if
$(\overline{u},\overline{v})\in H$. Motivated by the Tur\'an problem for
bipartite graphs, we say that $H\cap (X\times Y)$ is $(s,t)$-grid-free provided
that $G$ contains no complete bipartite subgraph that has $s$ vertices in $X$
and $t$ vertices in $Y$. We conjecture that every $(s,t)$-grid-free
hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree
in $\overline{y}$ is bounded by a constant $d = d(s,t)$, and we discuss
possible notions of the equivalence.
  We establish the result that if $H\cap(X\times \mathbb{P}^2)$ is
$(2,2)$-grid-free, then there exists $F\in
\mathbb{C}[\overline{x},\overline{y}]$ of degree $\le 2$ in $\overline{y}$ such
that $H\cap(X\times \mathbb{P}^2) = \{F = 0\}\cap (X\times \mathbb{P}^2)$.
Finally, we transfer the result to algebraically closed fields of large
characteristic.
