  In this paper we measure how efficiently a finite simple group $G$ is
generated by its elements of order $p$, where $p$ is a fixed prime. This
measure, known as the $p$-width of $G$, is the minimal $k\in \mathbb{N}$ such
that any $g\in G$ can be written as a product of at most $k$ elements of order
$p$. Using primarily character theoretic methods, we sharply bound the
$p$-width of some low rank families of Lie type groups, as well as the simple
alternating and sporadic groups.
