  We prove a new effective Chebotarev density theorem for Galois extensions
$L/\mathbb{Q}$ that allows one to count small primes (even as small as an
arbitrarily small power of the discriminant of $L$); this theorem holds for the
Galois closures of "almost all" number fields that lie in an appropriate family
of field extensions. Previously, applying Chebotarev in such small ranges
required assuming the Generalized Riemann Hypothesis. The error term in this
new Chebotarev density theorem also avoids the effect of an exceptional zero of
the Dedekind zeta function of $L$, without assuming GRH. We give many different
"appropriate families," including families of arbitrarily large degree. To do
this, we first prove a new effective Chebotarev density theorem that requires a
zero-free region of the Dedekind zeta function. Then we prove that almost all
number fields in our families yield such a zero-free region. The innovation
that allows us to achieve this is a delicate new method for controlling zeroes
of certain families of non-cuspidal $L$-functions. This builds on, and greatly
generalizes the applicability of, work of Kowalski and Michel on the average
density of zeroes of a family of cuspidal $L$-functions. A surprising feature
of this new method, which we expect will have independent interest, is that we
control the number of zeroes in the family of $L$-functions by bounding the
number of certain associated fields with fixed discriminant. As an application
of the new Chebotarev density theorem, we prove the first nontrivial upper
bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$,
applicable to infinite families of fields of arbitrarily large degree.
