  We consider a generic modified logarithmic Sobolev inequality (mLSI) of the
form $\mathrm{Ent}_{\mu}(e^f) \le \tfrac{\rho}{2} \mathbb{E}_\mu e^f
\Gamma(f)^2$ for some difference operator $\Gamma$, and show how it implies
two-level concentration inequalities akin to the Hanson--Wright or Bernstein
inequality. This can be applied to the continuous (e.\,g. the sphere or bounded
perturbations of product measures) as well as discrete setting (the symmetric
group, finite measures satisfying an approximate tensorization property,
\ldots).
  Moreover, we use modified logarithmic Sobolev inequalities on the symmetric
group $S_n$ and for slices of the hypercube to prove Talagrand's convex
distance inequality, and provide concentration inequalities for locally
Lipschitz functions on $S_n$. Some examples of known statistics are worked out,
for which we obtain the correct order of fluctuations, which is consistent with
central limit theorems.
