  In this article we study two "strong" topologies for spaces of smooth
functions from a finite-dimensional manifold to a (possibly
infinite-dimensional) manifold modeled on a locally convex space. Namely, we
construct Whitney type topologies for these spaces and a certain refinement
corresponding to Michor's $\mathcal{FD}$-topology. Then we establish the
continuity of certain mappings between spaces of smooth mappings, e.g.\ the
continuity of the joint composition map. As a first application we prove that
the bisection group of an arbitrary Lie groupoid (with finite-dimensional base)
is a topological group (with respect to these topologies). For the reader's
convenience the article includes also a proof of the folklore fact that the
Whitney topologies defined via jet bundles coincide with the ones defined via
local charts.
