Spatial confinement is known to induce a drastic change in the viscosity, relaxation times, and flow profile of liquids near the glass (or jamming) transition point. The essential underlying question is how a wall affects the dynamics of densely packed systems. Here we study this fundamental problem, using experiments on a driven granular hard-sphere liquid and numerical simulations of polydisperse and bidisperse colloidal liquids. The nearly hard-core nature of the particle-wall interaction provides an ideal opportunity to study purely geometrical confinement effects. We reveal that the slower dynamics near a wall is induced by wall-induced enhancement of `glassy structural order', which is a manifestation of strong interparticle correlations. By generalizing the structure-dynamics relation for bulk systems, we find a quantitative relation between the structural relaxation time at a certain distance from a wall and the correlation length of glassy structural order there. Our finding suggests that glassy structural ordering may be the origin of the slow glassy dynamics of a supercooled liquid.