Extremal density for subdivisions with length or sparsity constraints

Given a graph $H$, a balanced subdivision of $H$ is obtained by replacing all edges of $H$ with internally disjoint paths of the same length. In this paper, we prove that for any graph $H$, a linear-in-$e(H)$ bound on average degree guarantees a balanced $H$-subdivision. This strengthens an old result of Bollob\'as and Thomason, and resolves a question of Gil-Fern\'{a}ndez, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever $H$ is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs $H$ below this density, its subdivisions are forcible by a sublinear-in-$e(H)$ bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph $H$ with sublogarithmic density are forcible by a sublinear-in-$e(H)$ bound on average degree, provided that $H$ satisfies some additional separability condition.