Let P s be the s -dimensional complex projective space, and let X , Y be two non-empty open subsets of P s in the Zariski topology. A hypersurface H in P s × 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 u ¯ ∼ v ¯ if and only if ( u ¯ , v ¯ ) ∈ H . Motivated by the Turán problem for bipartite graphs, we say that H ∩ ( X × 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 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 ∩ ( X × P 2 ) is ( 2 , 2 ) -grid-free, then there exists F ∈ C [ x ¯ , y ¯ ] of degree ≤ 2 in y ¯ such that H ∩ ( X × P 2 ) = { F = 0 } ∩ ( X × P 2 ) . Finally, we transfer the result to algebraically closed fields of large characteristic. [ABSTRACT FROM AUTHOR]