Over the past decade, the ability to reduce the dimensions of fluidic devices to the nanometre scale (by using nanotubes(1-5) or nanopores(6-11), for example) has led to the discovery of unexpected water- and ion-transport phenomena(12-14). More recently, van der Waals assembly of two-dimensional materials(15) has allowed the creation of artificial channels with angstrom-scale precision(16). Such channels push fluid confinement to the molecular scale, wherein the limits of continuum transport equations(17) are challenged. Water films on this scale can rearrange into one or two layers with strongly suppressed dielectric permittivity(18,19) or form a room-temperature ice phase(20). Ionic motion in such confined channels(21) is affected by direct interactions between the channel walls and the hydration shells of the ions, and water transport becomes strongly dependent on the channel wall material(22). We explore how water and ionic transport are coupled in such confinement. Here we report measurements of ionic fluid transport through molecular-sized slit-like channels. The transport, driven by pressure and by an applied electric field, reveals a transistor-like electrohydrodynamic effect. An applied bias of a fraction of a volt increases the measured pressure-driven ionic transport (characterized by streaming mobilities) by up to 20 times. This gating effect is observed in both graphite and hexagonal boron nitride channels but exhibits marked material-dependent differences. We use a modified continuum framework accounting for the material-dependent frictional interaction of water molecules, ions and the confining surfaces to explain the differences observed between channels made of graphene and hexagonal boron nitride. This highly nonlinear gating of fluid transport under molecular-scale confinement may offer new routes to control molecular and ion transport, and to explore electromechanical couplings that may have a role in recently discovered mechanosensitive ionic channels(23).