Concentric-tube robots can offer a suitable compromise between force and curvature control. In a previous study by the authors, a real-time trajectory tracking scheme for an unloaded concentric-tube robot was developed. One of the practical barriers to the use of a concentric-tube robot in medical applications is compensation for the impact of environmental forces which can cause drastic deterioration in tracking performance. In this paper, by modifying the robot's forward kinematics and Jacobian, a new method is developed to facilitate tip tracking in real-time while accounting for an external load at the robot's tip. By considering the tip deflection resulting from the external load, a novel dual-layer control architecture is proposed to compensate for this deflection during trajectory tracking. In order to measure the force exerted on the tip position of the robot, a new technique is proposed that can move the sensing system from the distal tip to the proximal base. Experimental results are given to illustrate the effectiveness of the proposed method.