Experimental results are presented for the influence of Reynolds number on multifractal scale similarity in turbulent flows. These are obtained from single-point measurements of a dynamically passive Sc≈1 conserved scalar quantity ζ(t) in a turbulent shear flow at outer-scale Reynolds numbers of 14000[les ]Reδ[les ]110000. Statistical criteria based on the maximum allowable scale-to-scale variation L1(ε) in multiplier distributions P(Mε) from multifractal gauge sets allow accurate discrimination between multifractal and non-multifractal scaling. Results show that the surrogate scalar energy dissipation rate χs(t)≡(dζ/dt)2is found to display a scale-invariant similarity consistent with a random multiplicative cascade characterized by a bilinear multiplier distribution P(Mε) over a range of scales extending downward from the outer scaleTδ. For a range of scales extending upward from the inner (diffusive) scale TD, the dissipation rate displays a different scale-invariant similarity characterized by a uniform multiplier distribution. The former scale-invariance becomes evident in the present Sc≈1 data only when Reδ is sufficiently large. Comparisons with results from Sc 1 data indicate that this scale-invariant similarity applies when the outer-to-inner scale ratio Tδ/TD≈0.09 Re3/4δSc1/2 is greater than about 400. In contrast to the scalar dissipation rate field, the scalar field is found to lack any multifractal scale similarity.