Recent experimental studies have revealed unconventional phase and transport behaviors of water confined within lamellar graphene oxide membranes, which hold great promise not only in improving our current understanding of nanoconfined water but also in developing high-performance filtration and separation applications. In this work, we explore molecular structures and diffusive dynamics of water intercalated between graphene or graphene oxide sheets. We identify the monolayer structured water between graphene sheets at temperature T below Tc = ∼315 K and an interlayer distance d = 0.65 nm, which is absent as the sheets are oxidized. The non-continuum collective diffusion of water intercalation between graphene layers facilitates fast molecular transport due to reduced wall friction. This solid-like structural order of intercalated water is disturbed as T or d increases to a critical value, with abnormal declines in the coefficients of collective diffusion. Based on a patched model of graphene oxide sheets consisting of spatially distributed pristine and oxidized regions, we conclude that the non-continuum collective diffusion of intercalated water can explain fast water permeation through graphene oxide membranes as reported in recent experimental studies, in stark contrast to the conventional picture of pressure-driven continuum flow with boundary slip, which has been widely adopted in literature but may apply only at high humidity or in the fully hydrated conditions.