  We extend the work of Salberger; Walsh; Castryck, Cluckers, Dittmann and
Nguyen; and Vermeulen to prove the uniform dimension growth conjecture of
Heath-Brown and Serre for varieties of degree at least $4$ over global fields.
As an intermediate step, we generalize the bounds of Bombieri and Pila to
curves over global fields and in doing so we improve the $B^{\varepsilon}$
factor by a $\log(B)$ factor.
