A flexible needle can be accurately steered by robotically controlling the bevel tip orientation as the needle is inserted into tissue. Friction between the long, flexible needle shaft and the tissue can cause a significant discrepancy between the orientation of the needle tip and the orientation of the base where the needle angle is controlled. Our experiments show that several common phantom tissues used in needle steering experiments impart substantial friction forces to the needle shaft, resulting in a lag of more than 45 ( degrees ) for a 10 cm insertion depth in some phantoms; clinical studies report torques large enough to cause similar errors during needle insertions. Such angle discrepancies will result in poor performance or failure of path planners and image-guided controllers, since the needles used in percutaneous procedures are too small for state-of-the-art imaging to accurately measure the tip angle. To compensate for the angle discrepancy, we develop an estimator using a mechanics-based model of the rotational dynamics of a needle being inserted into tissue. Compared to controllers that assume a rigid needle in a frictionless environment, our estimator-based controller improves the tip angle convergence time by nearly 50% and reduces the path deviation of the needle by 70%.