Given a graph F, let st
(F) be the number of subdivisions of F, each with a different vertex set, which one can guarantee in a graph G in which every edge lies in at least t copies of F. In 1990, Tuza asked for which graphs F and large t, one has that st
(F) is exponential in a power of t. We show that, somewhat surprisingly, the only such F are complete graphs, and for every F which is not complete, st
(F) is polynomial in t. Further, for a natural strengthening of the local condition above, we also characterize those F for which st
(F) is exponential in a power of t.