  We initiate the systematic study of the following Tur\'an-type question.
Suppose $\Gamma$ is a graph with $n$ vertices such that the edge density
between any pair of subsets of vertices of size at least $t$ is at most $1 -
c$, for some $t$ and $c > 0$. What is the largest number of edges in a subgraph
$G \subseteq \Gamma$ which does not contain a fixed graph $H$ as an induced
subgraph or, more generally, which belongs to a hereditary property
$\mathcal{P}$? This provides a common generalization of two recently studied
cases, namely $\Gamma$ being a (pseudo-)random graph and a graph without a
large complete bipartite subgraph. We focus on the interesting case where $H$
is a bipartite graph.
  We determine the answer up to a constant factor with respect to $n$ and $t$,
for certain bipartite $H$ and for $\Gamma$ either a dense random graph or a
Paley graph with a square number of vertices. In particular, our bounds match
if $H$ is a tree, or if one part of $H$ has $d$ vertices complete to the other
part, all other vertices in that part have degree at most $d$, and the other
part has sufficiently many vertices. As applications of the latter result, we
answer a question of Alon, Krivelevich, and Samotij on the largest subgraph
with a hereditary property which misses a bipartite graph, and determine up to
a constant factor the largest number of edges in a string subgraph of $\Gamma$.
The proofs are based on a variant of the dependent random choice and a novel
approach for finding induced copies by inductively defining probability
distributions supported on induced copies of smaller subgraphs.
