A method for operating a wind turbine includes operating, via a controller, the wind turbine according to a speed set point during normal operation of the wind turbine. The method also includes receiving, via the controller, a command to shut down the wind turbine or to curtail operation of the wind turbine. In response to receiving the command, the method includes initiating, via the controller, a shutdown procedure or a curtailment procedure of the wind turbine. During the shutdown procedure or the curtailment procedure of the wind turbine, the method includes dynamically adjusting a rate of change of the speed set point as a function of a speed tracking error, which corresponds to a difference between an actual rotor speed of the wind turbine and the speed set point.