We prove that, given integers $l,q\geq 2$ and $n$ there exists an integer $\alpha$ such that, if $M$ is a simple matroid with no $l+2$-point line minor and at least $\alpha {q}^{r(M)}$ elements, then $M$ contains a $\mathrm{PG}(n-1,q')$-minor, for some prime-power $q'>q$.