A lubrication system is provided for a turbine engine. This lubrication system includes a lubricant source, a pump, a first turbine engine component, a bypass circuit and a second turbine engine component. The lubricant source includes a source outlet. The pump includes a pump inlet and a pump outlet. The pump inlet is fluidly coupled with the source outlet. The first turbine engine component includes a first volume. The first volume is fluidly coupled with the pump outlet. The bypass circuit includes a bypass inlet and a bypass outlet. The bypass inlet is fluidly coupled with the pump outlet upstream of the first volume. The bypass outlet is fluidly coupled with the pump inlet downstream of the source outlet. The second turbine engine component includes a second volume. The second volume is fluidly coupled with the pump inlet downstream of the bypass outlet.