Decaying quasi-two-dimensional turbulence in a thin-layer flow is explored in laboratory experiments. We report the presence of power-law interval in the enstrophy decay law, in agreement with earlier experiments by Cardoso et al. [Phys. Rev. E 49, 454 (1994)] and Hansen et al. [Phys. Rev. E 58, 7261 (1998)]. The decay exponent proves sensitive to the way in which the energy decay is compensated. For the range of initial microscale Reynolds numbers between 35 and 95, the decay exponent is close to -0.4 for the ratio of enstrophy to energy, and to -0.75 for the enstrophy multiplied with a compensating factor of exp(-2lambda(t)), where lambda is the bottom-drag coefficient and t the decay time. The vorticity behavior does not comply with the theory of Carnevale et al. [Phys. Rev. Lett. 66, 2735 (1991)]: robust vortices are not observed in the vorticity field and the vorticity kurtosis is less than the Gaussian value.