  For every finite quasisimple group of Lie type $G$, every irreducible
character $\chi$ of $G$, and every element $g$ of $G$, we give an exponential
upper bound for the character ratio $|\chi(g)|/\chi(1)$ with exponent linear in
$\log_{|G|} |g^G|$, or, equivalently, in the ratio of the support of $g$ to the
rank of $G$. We give several applications, including a proof of Thompson's
conjecture for all sufficiently large simple symplectic groups, orthogonal
groups in characteristic $2$, and some other infinite families of orthogonal
and unitary groups
