  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 bound in $e(H)$ 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.
