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This article is concerned with suppression of coupled axial–torsional vibration of drill-string system, taking into account the state-dependent delay inherited from regenerative cutting of drill-bit on rock and actuator saturation of drill-rig. Note that, in practice, the measurement and saturated control input could only be applied at the surface side of drill-string, while a nonlinearity of time-delay is introduced due to the bit-rock interaction at the downhole side. The main contribution of this article lies in three points. First, to capture the rich dynamics of drill-string, a multi-degree-of-freedom lumped parameter model of two differential equations is established for the axial and torsional motions of drill-string. A bit-rock interaction model for coupled cutting-frictional contact process is employed, resulting in a constraint on current and delayed states of drill-bit. Second, combining the drill-string and bit-rock interaction models, the dimensionless dynamic of coupled axial–torsional vibration of drill-string is described by a state-space equation of time-delay system with saturation nonlinearity. Third, using a second-order Bessel-Legendre integral inequality and a reciprocally convex inequality, delay-dependent sufficient conditions are provided to design a proper dynamic output-feedback controller with an antiwindup compensator for vibration suppression, using surface measurement and control only. The effectiveness of the approach is demonstrated through a case study of real data and an experiment based on a laboratory drill-rig.