  A folklore conjecture asserts the existence of a constant $c_n > 0$ such that
$\#\mathcal{F}_n(X) \sim c_n X$ as $X\to \infty$, where $\mathcal{F}_n(X)$ is
the set of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by
$X$. This conjecture is known if $n \leq 5$, but even the weaker conjecture
that there exists an absolute constant $C\geq 1$ such that $\#\mathcal{F}_n(X)
\ll_n X^C$ remains unknown and apparently out of reach. Here, we make progress
on this weaker conjecture (which we term the "uniform exponent conjecture") in
two ways. First, we reduce the general problem to that of studying relative
extensions of number fields whose Galois group is an almost simple group in a
natural primitive permutation representation. Second, for almost all such
groups, we prove the strongest known upper bound on the number of such
extensions. These bounds have the effect of resolving the uniform exponent
conjecture for solvable groups, sporadic groups, exceptional groups, and
classical groups of bounded rank.
