We propose and verify a wave-vector-space version of generalized extended self-similarity [R. Benzi et al., Europhys. Lett. 32, 709 (1995)] and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order ( $\le 20$) structure functions numerically for (1) the three-dimensional, incompressible Navier-Stokes equation (with and without hyperviscosity) and (2) the Gledzer-Ohkitani-Yamada shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers $4\times {10}^4\le {\mathrm{Re}}_{\lambda }\le 3\times {10}^6$, we find that the inertial-range exponents ( ${\zeta }_p$) of the order- $p$ structure functions do not approach their Kolmogorov value $p/3$ as ${\mathrm{Re}}_{\lambda }$ increases.

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