  The defective chromatic number of a graph class $\mathcal{G}$ is the minimum
integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is
$k$-colourable such that each monochromatic component has maximum degree at
most $d$. Similarly, the clustered chromatic number of a graph class
$\mathcal{G}$ is the minimum integer $k$ such that for some integer $c$, every
graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component
has at most $c$ vertices. This paper determines or establishes bounds on the
defective and clustered chromatic numbers of graphs with given girth in
minor-closed classes defined by the following parameters: Hadwiger number,
treewidth, pathwidth, treedepth, circumference, and feedback vertex number. One
striking result is that for any integer $k$, for the class of triangle-free
graphs with treewidth $k$, the defective chromatic number, clustered chromatic
number and chromatic number are all equal. The same result holds for graphs
with treedepth $k$, and generalises for graphs with no $K_p$ subgraph. We also
show, via a result of K\"{u}hn and Osthus~[2003], that $K_t$-minor-free graphs
with girth $g\geq 5$ are properly $O(t^{c_g})$ colourable, where $c_g\in(0,1)$
with $c_g\to 0$, thus asymptotically improving on Hadwiger's Conjecture.
