what are the other possibilities to quickly renderer Bezier paths on GPU?
In-Situ tesselation of a set of control points into convex patches defines by a triangular hull using a tesselation and/or geometry shader. Then passing the curvature parameters to a fragment shader that performs a per-fragment test if the fragment is within the boundaries of each patch and discarding it otherwise.
If an approximation is in order then just tesselating into a triangular mesh is in order.