  We address Heath-Brown's and Serre's dimension growth conjecture (proved by
Salberger), when the degree $d$ grows. Recall that Salberger's dimension growth
results give bounds of the form $O_{X, \varepsilon} (B^{\dim X+\varepsilon})$
for the number of rational points of height at most $B$ on any integral
subvariety $X$ of ${\mathbb P}^n_{\mathbb Q}$ of degree $d\geq 2$, where one
can write $O_{d,n, \varepsilon}$ instead of $O_{X, \varepsilon}$ as soon as
$d\geq 4$. Our main contribution is to remove the factor $B^\varepsilon$ as
soon as $d \geq 5$, without introducing a factor $\log B$, while moreover
obtaining polynomial dependence on $d$ of the implied constant. Working
polynomially in $d$ allows us to give a self-contained and slightly simplified
treatment of dimension growth for degree $d \geq 16$, while in the range $5
\leq d \leq 15$ we invoke results by Browning, Heath-Brown and Salberger. Along
the way we improve the well-known bounds due to Bombieri and Pila on the number
of integral points of bounded height on affine curves and those by Walsh on the
number of rational points of bounded height on projective curves. The former
improvement leads to a slight sharpening of a recent estimate due to Bhargava,
Shankar, Taniguchi, Thorne, Tsimerman and Zhao on the size of the $2$-torsion
subgroup of the class group of a degree $d$ number field. Our treatment builds
on recent work by Salberger which brings in many primes in Heath-Brown's
variant of the determinant method, and on recent work by Walsh and
Ellenberg--Venkatesh, who bring in the size of the defining polynomial. We also
obtain lower bounds showing that one cannot do better than polynomial
dependence on $d$.
