We measure wavefront (WF) distortions in a high-power thin-disk laser induced simultaneously by the gas-lens/wedge and disk front-surface deformation using a two-channel moiré-based WF sensor. Thermal lensing and tilting effects are characterized versus time, their pressure-dependent fluctuations are quantified, and finally the corresponding pure disk front-surface effects are estimated for zero pressure. A divergent probe beam with a WF mean curvature similar to the curvature of the disk is reflected off the disk front surface. The temporal evolution of the WF at laser start-up is characterized using the WF sensor. A camera records both temporal evolution of the moiré fringe patterns and the intensity profile of the laser beam to have simultaneously both phase and intensity profiles of the laser beam. Successive WF data quantify the temporal evolutions of the dioptric power and induced wedges in two directions and their fluctuations during laser operation. We investigate the effect of air pressure on the produced gas-lens/wedge. The method is not sensitive to translational vibrations and is very low cost with adjustable sensitivity.