  Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs.
We prove that if for some fixed $0<\delta\le 1$, every $n$-vertex graph of
$\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any
depth-$k$ minor (i.e. minor obtained by contracting disjoint subgraphs of
radius at most $k$) of a graph in $\mathcal{C}$ has average degree $O\big((k
\text{ polylog }k)^{1/\delta}\big)$. This confirms a conjecture of Dvo\v{r}\'ak
and Norin.
