Extremal density for sparse minors and subdivisions

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, • (1+o(1))t² average degree is sufficient to force the t× t grid as a topological minor; • $(3/2+o(1))t$ average degree forces everyt-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; • a universal bound of $(2+o(1))t$ on average degree forcing everyt-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.