A new trajectory fast tracking control strategy is proposed for quadcopter unmanned aerial vehicle (UAV) systems with suspended loads and specified output constraints. Firstly, a fixed-time disturbance barrier designed to estimate unknown nonlinear disturbances in real time, and a newly barrier function is constructed, from which the prescribed constrain performance of the loads is guaranteed in view of the barrier Lyapunov function method. Then, a novel nonsingular terminal sliding mode control law is developed, which not only avoids the occurrence of system singularity issue, but also ensures the fixed-time reachability of the sliding surface and the desirable property of the tracking errors. Finally, simulation results verify the effectiveness of the designed control method.