  Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse
subgraphs keeping the treewidth large) provided by the celebrated Grid Minor
theorem of Robertson and Seymour [JCTB '86] or by a classic result of Chekuri
and Chuzhoy [SODA '15], we show that for any natural numbers $t$ and $w$, and
real $\varepsilon > 0$, there is an integer $W := W(t,w,\varepsilon)$ such that
every graph with treewidth at least $W$ and no $K_{t,t}$ subgraph admits a
2-connected $n$-vertex induced subgraph with treewidth at least $w$ and at most
$(1+\varepsilon)n$ edges. The induced subgraph is either a subdivided wall, or
its line graph, or a spanning supergraph of a subdivided biclique. This in
particular extends a result of Weissauer [JCTB '19] that graphs of large
treewidth have a large biclique subgraph or a long induced cycle.
