  For a set of matroids $\mathcal{M}$, let $ex_\mathcal{M}(n)$ be the maximum
size of a simple rank-$n$ matroid in $\mathcal{M}$. We prove that, for any
finite field $\mathbb{F}$, if $\mathcal{M}$ is a minor-closed class of
$\mathbb{F}$-representable matroids of bounded branch-width, then $\lim_{n
\rightarrow \infty} ex_\mathcal{M}(n) / n$ exists and is a rational number,
$\Delta$. We also show that $ex_\mathcal{M}(n) - \Delta n$ is periodic when $n$
is sufficiently large and that $ex_\mathcal{M}$ is achieved by a subclass of
$\mathcal{M}$ of bounded path-width.
