The viscosity of hydrogen mixtures in the supercritical region of water is significant for the optimization of the gasification process of organic substances in supercritical water. In this work, a capillary apparatus was developed for measuring the viscosity of fluids and their mixtures at 279-980 K and up to 34 MPa. The viscosities of pure water, nitrogen, and hydrogen at 279-980 K and 0.5-31.0 MPa were measured, and the results were compared with the reference equations. Then, the viscosities of the H-2-CO2 and H-2-CO2-CH4 mixtures at 280-677 K and 0.7-33.1 MPa and of the H-2-H2O mixtures and the H-2-CO2-CH4-CO-H2O system at 720-924 K and 19.9-31.1 MPa were measured. The expanded relative uncertainty (k = 2) for the dynamic viscosity was estimated as 0.05. The viscosity data of the hydrogen mixtures were compared with those predicted using the equation of Wilke considering the nonideal gas effect. The absolute relative deviations between the calculated and experimental dynamic viscosities are less than 4.87%.