  Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all
$r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of
$\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of
degree $n$ $S_n$-extensions of $k$, improving upon results of Ellenberg, Pierce
and Wood as well as GRH-conditional results of Frei and Widmer. For large $r$,
our results are comparable with work of Heath-Brown and Pierce for imaginary
quadratic extensions of $\mathbb{Q}$. When $r=1$, our results are new even for
the family of all quadratic extensions of $\mathbb{Q}$, leading to an improved
upper bound for the count of degree $p$ $D_p$-extensions over $\mathbb{Q}$
(where $D_p$ is the dihedral group of order $2p$).
