  For any $\varepsilon > 0$, we give a polynomial-time
$n^\varepsilon$-approximation algorithm for Max Independent Set in graphs of
bounded twin-width given with an $O(1)$-sequence. This result is derived from
the following time-approximation trade-off: We establish an
$O(1)^{2^q-1}$-approximation algorithm running in time $\exp(O_q(n^{2^{-q}}))$,
for every integer $q \geqslant 0$. Guided by the same framework, we obtain
similar approximation algorithms for Min Coloring and Max Induced Matching. In
general graphs, all these problems are known to be highly inapproximable: for
any $\varepsilon > 0$, a polynomial-time $n^{1-\varepsilon}$-approximation for
any of them would imply that P$=$NP [Hastad, FOCS '96; Zuckerman, ToC '07;
Chalermsook et al., SODA '13]. We generalize the algorithms for Max Independent
Set and Max Induced Matching to the independent (induced) packing of any fixed
connected graph $H$. In contrast, we show that such approximation guarantees on
graphs of bounded twin-width given with an $O(1)$-sequence are very unlikely
for Min Independent Dominating Set, and somewhat unlikely for Longest Path and
Longest Induced Path. Regarding the existence of better approximation
algorithms, there is a (very) light evidence that the obtained approximation
factor of $n^\varepsilon$ for Max Independent Set may be best possible. This is
the first in-depth study of the approximability of problems in graphs of
bounded twin-width. Prior to this paper, essentially the only such result was
a~polynomial-time $O(1)$-approximation algorithm for Min Dominating Set [Bonnet
et al., ICALP '21].
