  For all positive integers $\ell$, we prove non-trivial bounds for the
$\ell$-torsion in the class group of $K$, which hold for almost all number
fields $K$ in certain families of cyclic extensions of arbitrarily large
degree. In particular, such bounds hold for almost all cyclic
degree-$p$-extensions of $F$, where $F$ is an arbitrary number field and $p$ is
any prime for which $F$ and the $p$-th cyclotomic field are linearly disjoint.
Along the way, we prove precise asymptotic counting results for the fields of
bounded discriminant in our families with prescribed splitting behavior at
finitely many primes.
