The glass-like slow dynamics in confluent epithelial monolayers is crucial for wound healing, embryogenesis, cancer progression, etc. Experiments have indicated several unusual properties in these systems. Unlike ordinary glasses, the glassiness in cellular systems strongly correlates with their static properties and is sub-Arrhenius. These results imply that the slow dynamics in epithelial monolayers is either not glassy or the underlying mechanism is different from ordinary glasses. Combining the analytical mode-coupling theory (MCT), vertex model simulations, and cellular experiments, we show that the slow dynamics is glassy, though the mechanism differs from ordinary glasses. The structure-dynamics feedback mechanism of MCT, and not the barrier-crossing mechanism, dominates the glassy dynamics, where the relaxation time diverges as a power law with a universal exponent 3/2 and naturally explains the sub-Arrhenius relaxation. Our results suggest the possibility of describing various complex biological processes like cell division and apoptosis via the static properties of the systems, such as average cell shape or shape variability.