Bleed air systems for use with aircrafts and related methods are disclosed. An example apparatus includes a turbo-compressor including a compressor has a compressor inlet fluidly coupled to a low-pressure compressor of an aircraft engine and an intermediate port of a high-pressure compressor of the aircraft engine. The compressor inlet to receive fluid from either the low-pressure compressor or the high-pressure compressor based on a first system parameter of the aircraft. A turbine has a turbine inlet fluidly coupled to the intermediate port of the high pressure compressor and a high-pressure port of the high pressure compressor of the aircraft engine. The turbine inlet to receive fluid from either the intermediate port of the high-pressure compressor or the high-pressure port of the high-pressure compressor based on a second system parameter of the aircraft.