Why is there no UV catastrophe (divergence) in turbulence?

It would help to know the context in which you read the phrase you have a problem understanding.

Regarding the scaling vs Reynolds problem, this part is pretty straighforward. The classical statistical theory of turbulence says that the energy cascades from the low wave number modes (typical scale L) to the high wave numbers modes (typical scale µ) where it is dissipated by viscosity.

In a statistical steady state the energy produced at L must be dissipated at µ. Now it is established that L/µ = Re^(3/4). So one clearly sees that as Reynolds increases, µ decreases and is 0 in the infinite Re limit. So this relation can indeed be described (imho misleadingly) by "separation of macro and micro increases".

From the dynamical point of view at infinite Re, the flow becomes inviscid and the Navier Stokes equations are substituted by the Euler equation.

The "UV catastrophy" is more mysterious. Formally one could say that as µ goes to 0, the energy density of the dissipative domain goes to infinity (UV divergence ?). However to that 2 remarks :

1) Navier Stokes is continuous but when µ arrives at molecular scales, the flow is no more continuous and Navier Stokes breaks down.

2) It is true that the spectral energy density increases at high wave numbers. But the nature shows us that there is actually no divergence - you will never see a very small vortex spinning infinitely fast.