Growth rate functions of dense classes of representable matroids

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