We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh–Bénard convection between free-slip boundaries. We focus on the ability of the convection to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers (Pr) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number (Ra) sufficiently, and we explore the resulting convection for Ra up to 1010. When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as Ra→∞. The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with Ra. When the large-scale shear is present with Pr≲2, the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with Ra for Pr=1. When the shear is present with Pr≳3, the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of Ra, but the growth rates are slower than any previously reported for Rayleigh–Bénard convection without large-scale shear. We find that the Nusselt numbers grow proportionally to Ra0.077 when Pr=3 and to Ra0.19 when Pr=10. Analogies with tokamak plasmas are described.