  We prove a homological stability theorem for families of discrete groups
(e.g. mapping class groups, automorphism groups of free groups, braid groups)
with coefficients in a sequence of irreducible algebraic representations of
arithmetic groups. The novelty is that the stable range is independent of the
choice of representation. Combined with earlier work of
Bergstr\"om--Diaconu--Petersen--Westerland this proves the
Conrey--Farmer--Keating--Rubinstein--Snaith predictions for all moments of the
family of quadratic $L$-functions over function fields, for sufficiently large
odd prime powers.
