Channels with a sinuous planform are common in both continental and deep-marine environments on Earth, and similarly in high-resolution images of the surface of Mars. Whereas common in rivers, continuous lateral channel migration and point-bar deposition appear to be much less common in the deep sea. In the bends of rivers, near-bed flow driving point bar growth results from a cross-flow superelevation of the water surface that sets up a lateral hydrostatic pressure gradient driving an inward-directed flow near the bed. However, in deep-marine systems the surface between the turbidity current and overlying ambient fluid sits well above the channel margins, and therefore precludes a similar lateral superelevation of the current top. Here it is argued that the cross-flow component is related to a density gradient that mimics the effect of the hydrostatic pressure gradient in rivers, and develops as coarse suspended particles that experience little uplift, and therefore negligible overspill, become concentrated along the outer bank. This condition develops best in a two-part suspension made up of a highly concentrated, unstratified basal plug of coarse sediment overlain sharply by a dilute cloud of much finer sediment—a density structure that differs from the more typical upward exponential decrease in density. The abundance of coarse and fine sand, but depletion in the intermediate grain size fraction, is related to transgressive shelf processes and its influence on sediment supplied to the system, and in turn, the flow structure of the current. It is under these seemingly uncommon granulometric conditions that continuous laterally migrating channels, and accordingly, riverine-like point-bar deposition, is most common in the deep sea.