We present a new method to upper bound 2-Selmer structures in families where one twists by the value of any integer polynomial P in any number of variables. From this one deduces that (for several elliptic curves E over Q) the families of elliptic curves EP(v) have bounded rank in average as v ranges over integer points in an expanding box. Similarly, this method provides an upper bound of the expected order of magnitude for the average 12-torsion of the class group of Q( D), as D ranges among squarefree integers. This is joint work with P. Koymans and E. Sofos.