  For each proper minor-closed subclass $\cM$ of the $\GF(q^2)$-representable
matroids containing all simple $\GF(q)$-representable matroids, we give, for
all large $r$, a tight upper bound on the number of points in a rank-$r$
matroid in $\cM$, and construct a rank-$r$ matroid in $\cM$ for which equality
holds. As a consequence, we give a tight upper bound on the number of points in
a $\GF(q^2)$-representable, rank-$r$ matroid with no $\PG(k,q^2)$-minor.
