  We prove an asymptotically tight bound on the extremal density guaranteeing
subdivisions of bounded-degree bipartite graphs with a mild separability
condition. As corollaries, we answer several questions of Reed and Wood on
embedding sparse minors. Among others,
  $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$
grid as a topological minor;
  $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph
as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the
value is the same for $t$-vertex graphs embeddable on any fixed surface;
  $\bullet$ a universal bound of $(2+o(1))t$ on average degree forcing every
$t$-vertex graph in any nontrivial minor-closed family as a minor, and the
constant 2 is best possible by considering graphs with given treewidth.
