In the design of offshore wind turbines, it is important to make a realistic estimate of the wind load. This is particularly important for floating wind turbines, having natural frequencies in a frequency range where the wind loads are high and large turbulent structures exist. This study shows that turbulence modelling greatly impacts the response of a 15-MW floating wind turbine. The turbulence models recommended by the International Electrotechnical Commission (IEC) are challenged by considering two additional models: Large Eddy Simulations (LES) and an approach using input from offshore wind measurements (TIMESR). The two standard models, the Kaimal spectrum with IEC coherence model (Kaimal) and the Mann spectral tensor model (Mann), differ in their coherence formulation. This results in higher standard deviations for the surge and pitch motions, and lower for the yaw motion, when applying Kaimal in comparison to Mann. For the specific floater of this study, more damage is obtained in the mooring lines when applying Kaimal. Applying the more realistic models, LES and TIMESR, increases the range of response further, concluding that the two standard turbulence models may lead to incorrect estimations of the response of a floating wind turbine. LES and TIMESR take atmospheric stability into account, which is proven to alter the response significantly.