jats:pWe present an experimental and theoretical study of the dynamics of laterally confined marine ice sheets in the natural limit in which the long, narrow channel into which they flow is wider than the depth of the ice. A marine ice sheet comprises a grounded ice sheet in contact with bedrock that floats away from the bedrock at a ‘grounding line’ to form a floating ice shelf. We model the grounded ice sheet as a viscous gravity current resisted dominantly by vertical shear stresses owing to the no-slip boundary condition applied at the bedrock. We model the ice shelf as a floating viscous current resisted dominantly by horizontal shear stresses owing to no-slip boundary conditions applied at the sidewalls of the channel. The two shear-dominated regions are coupled by jump conditions relating force and fluid flux across a short transition region downstream of the grounding line. We find that the influence of the stresses within the transition region becomes negligible at long times and we model the transition region as a singular interface across which the ice thickness and mass flux can be discontinuous. The confined shelf buttresses the sheet, causing the grounding line to advance more than it would otherwise. In the case that the sheet flows on a base of uniform slope, we find asymptotically that the grounding line advances indefinitely as jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016000379_inline1" />jats:tex-math$t1/3$</jats:tex-math></jats:alternatives></jats:inline-formula>, where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016000379_inline2" />jats:tex-math$t$</jats:tex-math></jats:alternatives></jats:inline-formula> is time. This contrasts with the two-dimensional counterpart, for which the shelf provides no buttressing and the grounding line reaches a steady state (Robison, jats:italicJ. Fluid Mech.</jats:italic>, vol. 648, 2010, pp. 363–380).</jats:p>