In this paper, the trajectory tracking control of a quadrotor UAV system with multi-source disturbances is studied. A full loop composite fast nonsingular terminal sliding mode trajectory tracking control scheme is proposed by considering both position and attitude loop dynamics. Firstly, the trajectory tracking problem is transformed into the commands tracking in position and attitude loop. Then, the couplings between different channels and the influences of multi-source disturbances are regarded as lumped disturbances and estimated by extended state observers. Based on the disturbances' estimation and the fast nonsingular terminal sliding mode control algorithm, the composite fast nonsingular terminal sliding mode controllers are constructed both in the position loop and the attitude loop. Finally, the real control action of UAV, i.e., rotor speed, is calculated based on the virtual controller. The simulation results show that the proposed control scheme significantly improves the trajectory tracking performance.