Electronic devices with sliding device housings and translating flexible displays and corresponding methods

An electronic device includes a flexible display and a device housing that defines a translation surface for the flexible display. A support layer is positioned between a portion of the flexible display and the translation surface. A rotor positioned within a curvilinear section of the flexible display causes a linear translation of the support layer across the translation surface by drawing the flexible substrate around the rotor. A flexible support structure can be positioned between the support layer and the flexible display, with the flexible support structure defining a plurality of engagement teeth with ends extending under one or more housing rails to prevent pillowing of the flexible display.