Membrane-based separations have been increasingly utilized to address global energy crisis and water scarcity. However, the separation efficiency often suffers from the trade-off between membrane permeability and selectivity. Although great efforts have been devoted, a membrane with both high permeability and high selectivity remains a distant prospect. Inspired by the hourglass structure and ultrafast water transport in aquaporins, we propose a novel approach to fabricating membranes with conical nanochannels to reduce the mass transfer resistance and to introduce Laplace pressure as the internal driving force, which successfully breaks the permeability/selectivity trade-off. First, sulfonated polyaniline (SPANI) nanorods were in situ-synthesized and vertically aligned on sulfonated graphene oxide (SGO) nanosheets, forming SGO-SPANI X composites. Then, the graphene oxide (GO) membranes were fabricated by assembling SGO-SPANI X composites through pressure-assisted filtration, in which the SPANI nanorods would bend and flatten on the SGO nanosheets under low shear force, forming stripe arrays on SGO nanosheets. The tilted stripe arrays between the adjacent SGO nanosheets form the conical nanochannels inside GO membranes. The conical nanochannels significantly decreased the steric hindrance and enabled the generation of Laplace pressure as the internal driving force within membranes. Consequently, the resulting membranes exhibit an ultrahigh water permeability of 1222.77 L·m-2·h-1·bar-1 and high efficiency in dye removal from water with a rejection of 90.44% and permeability of 528 L·m-2·h-1·bar-1.