  The "extremal function" $c(H)$ of a graph $H$ is the supremum of densities of
graphs not containing $H$ as a minor, where the "density" of a graph $G$ is the
ratio of the number of edges to the number of vertices. Myers and Thomason
(2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019)
determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs
$H$, as well as almost all graphs $H$ of constant density.
  We explore the asymptotic behavior of the extremal function in the regime not
covered by the above results, where in addition to having constant density the
graph $H$ is in a graph class admitting strongly sublinear separators. We
establish asymptotically tight bounds in many cases. For example, we prove that
for every planar graph $H$, $$c(H) =
(1+o(1))\cdot\max\left\{\frac{|V(H)|}{2},|V(H)| - \alpha (H)\right\},$$
extending recent results of Haslegrave, Kim and Liu (2020). We also show that
an asymptotically tight bound on the extremal function of graphs in
minor-closed families proposed by Haslegrave, Kim and Liu (2020) is equivalent
to a well studied open weakening of Hadwiger's conjecture.
