Dealing with torsional vibrations and stick–slip oscillations of a drill string system is a challenging engineering task in the oil drilling process because of the harmful and costly consequences of such vibrations. In this article, the drill string system is modeled using a lumped-parameter model with four degrees of freedom, and the bit–rock contact is represented by a nonlinear function of a bit velocity. Also, tracking the desired velocity of a drill string system with known constant input delay is addressed in the presence of external disturbance and parameter uncertainties by applying the Smith predictor–based sliding mode control method. The performance of the smith predictor–based sliding mode control with input delay and disturbance in tracking the desired velocity and controlling the stick–slip oscillations is compared with the sliding mode control with/without input delay. The system output’s sensitivity to the delay parameter is also investigated, indicating how the bit velocity changes concerning the delay parameter. The proper choice of adaptation gain is determinative in the performance of the controller, and its impact is investigated. Moreover, the robustness of the smith predictor–based sliding mode control is shown by changing the weight on the bit parameter. Simulation results demonstrate the effectiveness of the proposed method.