  We prove that for every $t\in \mathbb{N}$ there is a constant $\gamma_t$ such
that every graph with twin-width at most $t$ and clique number $\omega$ has
chromatic number bounded by $2^{\gamma_t \log^{4t+3} \omega}$. In other words,
we prove that graph classes of bounded twin-width are quasi-polynomially
$\chi$-bounded. This provides a significant step towards resolving the question
of Bonnet et al. [ICALP 2021] about whether they are polynomially
$\chi$-bounded.
