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The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertexgraph containing no copy of the complete bipartite graph $K_{s,s}$ has at most$O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applicationsin discrete geometry and structural graph theory, a number of resultsdemonstrated that this bound can be greatly improved if the graph satisfiescertain structural restrictions. We propose the systematic study of thisphenomenon, and state the conjecture that if $H$ is a bipartite graph, then aninduced $H$-free and $K_{s,s}$-free graph cannot have much more edges than an$H$-free graph. We provide evidence for this conjecture by considering trees,cycles, the cube graph, and bipartite graphs with degrees bounded by $k$ on oneside, obtaining in all the cases similar bounds as in the non-induced setting.Our results also have applications to the Erd\H{o}s-Hajnal conjecture, theproblem of finding induced $C_4$-free subgraphs with large degree and boundingthe average degree of $K_{s, s}$-free graphs which do not contain inducedsubdivisions of a fixed graph.

K\H{o}v\'ari-S\'os-Tur\'an theorem for hereditary families