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.