  Let $G$ be a finite classical group generated by transvections, i.e., one of
$\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$,
$\operatorname{Sp}_{2n}(q)$, or $\operatorname{O}^\pm_{2n}(q)$ ($q$ even), and
let $X$ be a generating set for $G$ containing at least one transvection.
Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of
the Cayley graph $\operatorname{Cay}(G, X)$ is bounded by $(n \log q)^C$ for
some constant $C$. This confirms Babai's conjecture on the diameter of finite
simple groups in the case of generating sets containing a transvection.
  By combining this with a result of the author and Jezernik it follows that if
$G$ is one of $\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$,
$\operatorname{Sp}_{2n}(q)$ and $X$ contains three random generators then with
high probability the diameter $\operatorname{Cay}(G, X)$ is bounded by
$n^{O(\log q)}$. This confirms Babai's conjecture for non-orthogonal classical
simple groups over small fields and three random generators.
