We probe the rheology of the model liquid octamethylcyclotetrasiloxane
(OMCTS) confined into molecularly thin films, using a unique Surface Forces
Apparatus allowing to explore a large range of shear rates and confinement. We
thus show that OMCTS under increasing confinement exhibits the viscosity
enhancement and the non-linear flow properties characteristic of a sheared
supercooled liquid approaching its glass transition. Besides, we study the
drainage of confined OMCTS via the propagation of "squeeze-out" fronts. The
hydrodynamic model proposed by Becker and Mugele [Phys. Rev. Lett. {\bf 91},
166104 (2003)] to describe such front dynamics leads to a conclusion in
apparent contradiction with the dynamical slowdown evidenced by rheology
measurements, which suggests that front propagation is not controlled by large
scale flow in the confined films