The main focus is the Reynolds number dependence of Kolmogorov normalized low-order moments of longitudinal and transverse velocity increments. The velocity increments are obtained in a large number of flows and over a wide range (40–4250) of the Taylor microscale Reynolds number Rλ. The Rλ dependence is examined for values of the separation, r, in the dissipative range, inertial range and in excess of the integral length scale. In each range, the Kolmogorov-normalized moments of longitudinal and transverse velocity increments increase with Rλ. The scaling exponents of both longitudinal and transverse velocity increments increase with Rλ, the increase being more significant for the latter than the former. As Rλ increases, the inequality between scaling exponents of longitudinal and transverse velocity increments diminishes, reflecting a reduced influence from the large-scale anisotropy or the mean shear on inertial range scales. At sufficiently large Rλ, inertial range exponents for the second-order moment of the pressure increment follow more closely those for the fourth-order moments of transverse velocity increments than the fourth-order moments of longitudinal velocity increments. Comparison with DNS data indicates that the magnitude and Rλ dependence of the mean square pressure gradient, based on the joint-Gaussian approximation, is incorrect. The validity of this approximation improves as r increases; when r exceeds the integral length scale, the Rλ dependence of the second-order pressure structure functions is in reasonable agreement with the result originally given by Batchelor (1951).