Why does a turbulent boundary layer cause more friction drag than a laminar boundary layer?
No, a higher Reynolds number only signifies a lower ratio of viscous to inertial forces. Since you increase the Reynolds number by either increasing speed or length, both will drive up the inertial forces, making the friction forces relatively lower.

Let's do a Gedankenexperiment: Consider an infinitely thin flat plate which moves through air in its lengthwise direction, so there is no pressure gradient along the flow path. The friction force caused by the boundary layer is F. Now add some length at the trailing edge of the plate. The added surface will add some more drag F', so the total drag will be F + F', both being positive. Since the length has increased, the Reynolds number of the flow has increased in the same way, but so has the friction force.

What has decreased is the coefficient of friction, however, because the thicker boundary layer towards the end of the flat plate will cause less friction force per unit of length. The coefficient is calculated by dividing the force by the dynamic pressure and the surface area, and since the surface area has increased by more than the drag, the coefficient of friction is lower.

You can repeat the experiment with a flow speed increase, and now the friction force will go up, too. So both ways to increase the Reynolds number will increase friction drag. Again, the drag increase is lower relative to the increase in dynamic pressure because the boundary layer at the higher speed is thinner, and again only the coefficient of friction is lower.

Only now, when that is out of the way, I want to answer your question directly: If you compare fully turbulent boundary layers, an increase in the Reynolds number will lower the coefficient of drag but will increase friction drag as outlined above.

Now compare two similar fully laminar flows at two different Reynolds numbers, and again the flow with the higher Reynolds number will experience more friction drag. The different speed profiles of their respective boundary layers will only be significant when you compare a laminar flow with a turbulent flow at the same or similar Reynolds number. In the range of Reynolds numbers typical for aircraft the turbulent flow will cause more friction drag.

If you now repeat the initial Gedankenexperiment with a laminar boundary layer and add some length at a point where the local Reynolds number is safely above 500,000, the local flow will already have transitioned to turbulent and the added length will add a fully turbulent portion. Now it can happen that the added surface will experience more drag per unit of area than the forward, mostly laminar surface. The result is both a higher friction drag and a higher coefficient of friction.

If you allow me to copy the illustration from this answer here, the effect can be easily seen:



Friction drag coefficient over chord for an E502mod airfoil at 3° AoA. Blue: Top surface, Red: Bottom surface.

Now this plot uses a real airfoil with a changing pressure coefficient over length, but with a flat plate the plot would look similar. The much higher local speed gradient at the surface of the boundary layer in the rear part will add much more drag per length and the total drag will increase disproportionately. However, a drag decrease with increasing Reynolds numbers is nowhere to be seen.