The control design of wheeled mobile robots is often accomplished based on the robot’s kinematics which imposes critical challenges in the motion tracking control of such systems. In the related literature, most works designed dynamic controllers based on the terms of voltage or torque; however, the velocity is often utilized in industrial and commercial applications. To address this issue, this paper focuses on the design of a velocity-based control for the trajectory-tracking problem of mobile robots in practical applications including measurement acquisition. The control design of the mobile robot is realized in two separate parts using kinematic control and dynamic control. The design of kinematic control is carried out based on the robot’s kinematics where the motion is described without considering the forces and torques. A radial basis function (RBF) neural network (NN) based on the model-free controller is designed for the dynamic controller of mobile robots without relying on the system dynamics. In the proposed dynamic control, a time-delay estimation is adopted to estimate the disturbances and noises in the mobile robot and remove them from the feedback loop control. Several typical scenarios of mobile robots with high-speed movements are conducted to assess the feasibility of the proposed NN controller-based time-delay estimation under noises and disturbances. To show the supremacy of the suggested controller, the dynamic responses of the mobile robot test system are compared with the prevalent controllers. The extensive simulation and real-time examinations reveal the capability of the proposed NN controller-based time-delay estimation to control high-speed mobile robots under high-level noises and disturbances.