  Dujmovi\'c et al. [\emph{J.~ACM}~'20] recently proved that every planar graph
is isomorphic to a subgraph of the strong product of a bounded treewidth graph
and a path. Analogous results were obtained for graphs of bounded Euler genus
or apex-minor-free graphs. These tools have been used to solve longstanding
problems on queue layouts, non-repetitive colouring, $p$-centered colouring,
and adjacency labelling. This paper proves analogous product structure theorems
for various non-minor-closed classes. One noteable example is $k$-planar graphs
(those with a drawing in the plane in which each edge is involved in at most
$k$ crossings). We prove that every $k$-planar graph is isomorphic to a
subgraph of the strong product of a graph of treewidth $O(k^5)$ and a path.
This is the first result of this type for a non-minor-closed class of graphs.
It implies, amongst other results, that $k$-planar graphs have non-repetitive
chromatic number upper-bounded by a function of $k$. All these results
generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a
more general setting based on so-called shortcut systems, which are of
independent interest. This leads to analogous results for certain types of map
graphs, string graphs, graph powers, and nearest neighbour graphs.
