  We prove a quantitative estimate with a power saving error term for the
number of filling closed geodesics of a given topological type and length $\leq
L$ on an arbitrary closed, orientable, negatively curved surface. More
generally, we prove estimates of the same kind for the number of free homotopy
classes of filling closed curves of a given topological type on a closed,
orientable surface whose geometric intersection number with respect to a given
filling geodesic current is $\leq L$. The proofs rely on recent progress made
on the study of the effective dynamics of the mapping class group on
Teichm\"uller space and the space of closed curves of a closed, orientable
surface, and introduce a novel method for addressing counting problems of
mapping class group orbits that naturally yields power saving error terms. This
method is also applied to study counting problems of mapping class group orbits
of Teichm\"uller space with respect to Thurston's asymmetric metric.
