This paper investigates the model based tracking control of soft growing robots with pneumatic actuation that extend according to the principle known as eversion. A model of the system which accounts for the pressure dynamics is presented. A new control law is constructed with a high-order sliding-mode approach and a nonlinear observer is employed to compensate for the effect of external forces. Numerical simulations and experiments demonstrate the effectiveness of the proposed controller compared to our former energy-shaping implementation and to a baseline sliding-mode controller. Experiments with a training phantom demonstrate that the new controller resulted in a reduced peak pressure, approximately 14.8% lower, a reduced tracking error, approximately 4.9% lower RMSE, and a reduced consumption of compressed air, approximately 3.9% lower, compared to a baseline sliding-mode algorithm.