Wave- and current-supported turbidity currents (WCSTCs) are one of the sediment delivery mechanisms from the inner shelf to the shelf break. Therefore, they play a significant role in the global cycles of geo-chemically important particulate matter. Recent observations suggest that WCSTCs can transform into self-driven turbidity currents close to the continental margin. However, little is known regarding the critical conditions that grow self-driven turbidity currents on WCSTCs. This is in part due to the knowledge gaps in the dynamics of WCSTCs regarding the role of density stratification. Especially the effect of sediment entrainment, and the parameters thereof, on density stratification and the amount of sediment suspension, has been overlooked. To this end, this study revisits the existing theoretical framework for a simplified WCSTC, in which waves are absent, i.e., alongshelf current-supported turbidity current (ACSTC). A depth-integrated advection model is developed for suspended sediment concentration. The analyses of the model, which are verified by turbulence-resolving simulations, indicate that the amount of suspended sediment load is regulated by the equilibrium among density stratification, positive feedback between entrainment and cross-shelf gravity force, and settling flux dissociated with density stratification. It is also found that critical density stratification is not a necessary condition for equilibrium. A quantitative relation is developed for the critical conditions for self-driven turbidity currents, which is a function of bed shear stress, entrainment parameters, bed slope, and sediment settling velocity. In addition, the suspended sediment load is analytically estimated from the model developed.