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