  The problem of finding dense induced bipartite subgraphs in $H$-free graphs
has a long history, and was posed 30 years ago by Erd\H{o}s, Faudree, Pach and
Spencer. In this paper, we obtain several results in this direction. First we
prove that any $K_t$-free graph with minimum degree at least $d$ contains an
induced bipartite subgraph of minimum degree at least $c_t \log d/\log \log d$,
confirming (asymptotically) several conjectures by Esperet, Kang and
Thomass\'e. Complementing this result, we further obtain optimal bounds for
this problem in the case of dense triangle-free graphs, and we also answer a
question of Erd\H{o}s, Janson, {\L}uczak and Spencer.
