Have a look at this:

Moody Diagram
Note how the on the right hand side of the diagram (high Reynolds number) the variation of friction coefficient with Reynolds number is almost flat.

I came across this which seems to address exactly the issues you are looking at:

Laminar_turbulent.pdf
Note particularly the Moody diagram on page 14.

Note in laminar flow there is a direct linear relationship between friction factor and Reynolds number.

In turbulent flow the relationship is much less direct, and in "fully" turbulent flow there is hardly any connection at all.