  The classical K\H{o}v\'ari-S\'os-Tur\'an theorem states that if $G$ is an
$n$-vertex graph with no copy of $K_{s,t}$ as a subgraph, then the number of
edges in $G$ is at most $O(n^{2-1/s})$. We prove that if one forbids $K_{s,t}$
as an induced/ subgraph, and also forbids any/ fixed graph $H$ as a (not
necessarily induced) subgraph, the same asymptotic upper bound still holds,
with different constant factors. This introduces a nontrivial angle from which
to generalize Tur\'an theory to induced forbidden subgraphs, which this paper
explores. Along the way, we derive a nontrivial upper bound on the number of
cliques of fixed order in a $K_r$-free graph with no induced copy of $K_{s,t}$.
This result is an induced analog of a recent theorem of Alon and Shikhelman and
is of independent interest.
