Small-dimensional representations of algebraic groups of type $A_l$

For $G$ an algebraic group of type $A_l$ over an algebraically closed field of characteristic $p$, we determine all irreducible rational representations of $G$ in defining characteristic with dimensions $\le (l+1{)}^s$ for $s=3,4$, provided that $l>18$, $l>35$ respectively. We also give explicit descriptions of the corresponding modules for $s=3$.