  We study the growth rate of weak coloring numbers of graphs excluding a fixed
graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017)
showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of
$X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to
a factor of $\mathcal{O}(r \log r)$. Moreover, we tie the exponent of the
polynomial to a structural property of $X$, namely, $2$-treedepth. As a result,
for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that
$\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$, which
improves on the bound $\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$
given by Dujmovi\'c et al. (SODA, 2024), where $g$ is an exponential function.
In the case of planar graphs of bounded treewidth, we show that the maximum
$r$-th weak coloring number is in $\mathcal{O}(r^2\mathrm{log}\ r$), which is
best possible.
