  In this article we prove several new uniform upper bounds on the number of
points of bounded height on varieties over $\mathbb{F}_q[t]$. For projective
curves, we prove the analogue of Walsh' result with polynomial dependence on
$q$ and the degree $d$ of the curve. For affine curves, this yields an
improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher
dimensions, we prove a version of dimension growth for hypersurfaces of degree
$d\geq 64$, building on work by Castryck, Cluckers, Dittmann and Nguyen in
characteristic zero. These bounds depend polynomially on $q$ and $d$, and it is
this dependence which simplifies the treatment of the dimension growth
conjecture.
