Horizontal convection (HC) serves as a canonical model for geophysical and industrial flows driven by differential heating along a surface. While the classical Oberbeck–Boussinesq (OB) approximation is well established, the impact of a nonlinear equation of state, specifically the density extremum of water near $4^{\circ }C$, remains underexplored. Here, we investigate this effect using two-dimensional numerical simulations over the Rayleigh-number range ${10}^6\le Ra\le 5\times {10}^{10}$. We examine four configurations, contrasting extremum (EXT) and monotonic (MON) buoyancy boundary conditions with linear (LE) and nonlinear (NE) equations of state. Our results reveal that, owing to strong cabbeling, the EXT-NE case develops mixing plumes that drive a strong bicellular circulation. At $Pr=1$, as $Re$ increases, this bicellular state becomes unstable and transitions to a single-roll circulation. This reorganization manifests as transitional anomalies in the Reynolds-number ($Re$) scaling, while the emergence of full-depth plumes alters the heat-transport mechanism. Consequently, in contrast to the classical Rossby scaling ($Nu\sim R{a}^{1/5}$) observed in the reference cases, the EXT-NE case exhibits enhanced heat-transport scaling ranging from $Nu\sim R{a}^{1/4}$ to $Nu\sim R{a}^{1/3}$. To interpret this behavior, we examine the total energy budget and identify an additional potential-energy transfer term, ${\Phi }_{i2}$, arising from the nonlinear equation of state. The scaling argument suggests that the magnitude of this contribution is controlled by the characteristic plume height ($\hat{z}$). Specifically, when the plumes penetrate the entire cavity depth ($\hat{z}\sim H$), as observed in the EXT-NE case, the global kinetic-energy dissipation is no longer described by the standard OB-HC energy closure alone. The resulting model captures the main trends of the numerical data and provides a possible energy-budget interpretation of the enhanced transport observed in this two-dimensional configuration.