We construct an adaptive controller for a linear minimum-phase system of an arbitrary relative degree with an unknown bounded disturbance and dynamically quantized measurements. The key novelty is the extension of the shunting method (parallel feedforward compensator) to plants with bounded disturbances. This method leads to an augmented system of relative degree one that is stabilized by a passification-based adaptive controller. Moreover, we design a switching procedure for the controller parameters and the quantizer's zoom that ensures the state convergence from an arbitrary set to an ellipsoid whose size depends on the disturbance bound. The results are demonstrated by an example of an aircraft flight control.