Uniform twisted homological stability

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 Bergström–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.