A method includes controlling a bearing fluid supply system to provide the bearing fluid to a hydrostatic bearing of the turbopump assembly. The bearing fluid includes a supercritical working fluid. The method also includes receiving data corresponding to a pressure of the bearing fluid measured at or near a bearing fluid drain fluidly coupled to the hydrostatic bearing, determining a thermodynamic state of the bearing fluid at or near the bearing fluid drain based at least in part on the received data, and controlling a backpressure regulation valve to throttle the backpressure regulation valve between an opened position and a closed position to regulate a backpressure in a bearing fluid discharge line to maintain the bearing fluid in a supercritical state in the hydrostatic bearing and/or at or near the bearing fluid drain.