In a horizontal convection (HC) system heat is supplied and removed exclusively through a single, top, or bottom, surface of a fluid layer. It is commonly agreed that in the studied Rayleigh number (Ra) range, the convective heat transport, measured by the Nusselt number, follows the Rossby (1965) scaling, which is based on the assumptions that the HC flows are laminar and determined by their boundary layers. However, the universality of this scaling is questionable, as these flows are observed to become more turbulent with increasing Ra. Here we propose a theoretical model for heat and momentum transport scalings with Ra, which is based on the Grossmann and Lohse (2000) ideas, applied to HC flows. The obtained multiple scaling regimes include in particular the Rossby scaling and the ultimate scaling by Siggers et al. (2004). Our results have bearing on the understanding of the convective processes in many geophysical systems and engineering applications.