  We give a lower bound for the delta invariant of the fundamental divisor of a
quasi-smooth weighted hypersurface. As a consequence, we prove K-stability of a
large class of quasi-smooth Fano hypersurfaces of index 1 and of all smooth
Fano weighted hypersurfaces of index 1 and 2. The proofs are based on the
Abban--Zhuang method and on the study of linear systems on flags of weighted
hypersurfaces.
