  We show that every graph with twin-width $t$ has chromatic number $O(\omega
^{k_t})$ for some integer $k_t$, where $\omega$ denotes the clique number. This
extends a quasi-polynomial bound from Pilipczuk and Soko{\l}owski and
generalizes a result for bounded clique-width graphs by Bonamy and Pilipczuk.
The proof uses the main ideas of the quasi-polynomial approach, with a
different treatment of the decomposition tree. In particular, we identify two
types of extensions of a class of graphs: the delayed-extension (which
preserves polynomial $\chi$-boundedness) and the right-extension (which
preserves polynomial $\chi$-boundedness under bounded twin-width condition).
Our main result is that every bounded twin-width graph is a delayed extension
of simpler classes of graphs, each expressed as a bounded union of right
extensions of lower twin-width graphs.
