  This paper shows that $K_t$-minor-free (and $K_{s, t}$-minor-free) graphs $G$
are subgraphs of products of a tree-like graph $H$ (of bounded treewidth) and a
complete graph $K_m$. Our results include optimal bounds on the treewidth of
$H$ and optimal bounds (to within a constant factor) on $m$ in terms of the
number of vertices of $G$ and the treewidth of $G$. These results follow from a
more general theorem whose corollaries include a strengthening of the
celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc.
1990] and the Planar Graph Product Structure Theorem of Dujmovi\'c et al. [J.
ACM 2020].
