Turbulence velocity measurements have been made in the surface layer of the atmosphere at Taylor microscale Reynolds numbers between 10,000 and 20,000. Even at these high Reynolds numbers, the structure functions do not scale unambiguously. It is shown that the scaling improves significantly by implementing a plausible correction due to the mean shear. For second and fourth order structure functions, the exponents for the corrected data are close to those determined by extended self-similarity (ESS). ESS improves scaling enormously for all orders, and is used to obtain exponents for moment orders between -0.08 and 10. Anomaly prevails even for very low orders. A major qualitative conclusion is that it is difficult to discuss the scaling effectively without first understanding quantitatively the effects of finite shear and finite Reynolds numbers.