NaughtyBezierCurves | Bezier Curve Game Object for Unity | Editor library

I want to dynamically draw 2D road segments in Unity. I know the 4 edge points of the segment and the required control points for the bezier curves. This allows me to draw the outer shape of the road segment pretty easily, wich results (in this example) in 2 end lines and 2 curves that form the shape of the road.

To draw the bezier curves, I am using this: https://github.com/dbrizov/NaughtyBezierCurves

This is what the outer shape looks like

The goal is to fill the road segment with some texture. How can I create some sort of planes or mesh that would have the same shape of the road?

My first idea was to go along both curves and create planes, maybe every 1/100 lenght of the curve, and add the texture to them. But with this approach, I dont know how to rotate the planes or how to prevent spaces or overlapping. Maybe someone has a simpler solution for this problem.

This is what I want to achive