  For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of
a connected Cayley graph of $G$. A well-known conjecture of Babai states that
$\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a
non-abelian finite simple group. Let $G$ be a finite simple group of Lie type
of Lie rank $n$ over the field $F_{q}$. Babai's conjecture has been verified in
case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently,
Biswas and Yang proved that $\mathrm{diam}(G)$ is bounded by $q^{O( n
{(\log_{2}n + \log_{2}q)}^{3})}$. We show that in fact $\mathrm{diam}(G) <
q^{O(n {(\log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller
than the order of $G$ for $n$ large, even if $q$ is large. As an application,
we show that more generally $\mathrm{diam}(H) < q^{O( n {(\log_{2}n)}^{2})}$
holds for any subgroup $H$ of $\mathrm{GL}(V)$, where $V$ is a vector space of
dimension $n$ defined over the field $F_q$.
