Bleed air systems for use with aircrafts and related methods are disclosed. An example apparatus includes a turbo-compressor including a compressor having a compressor inlet fluidly coupled to a low-pressure compressor of the aircraft engine and a compressor outlet fluidly coupled to a first system of an aircraft. The turbo-compressor also includes a turbine inlet fluidly coupled to a high-pressure compressor of the aircraft engine and a turbine outlet fluidly coupled to a second system of the aircraft.