Make a bezier curve into a subdivided mesh? I have created a shape using a bezier curve.  I want to give it depth and then apply an image as a displacement modifier.  

I have added a face to the shape and extruded the shape into 3 dimensions. However I cannot get the face to subdivide so I cannot apply the displacement modifier.

How can I get a detailed enough mesh across that face? 



The face I want to add detail to is selected.
For subdivide you need quads. Upper face is unsightly ngon and can't be subdivided. Remesh modifier should be helpful in this case.

If Remesh doesn't  do its job well, you can delete ngon and fill area manually with smaller parts by selecting two opposite verts and pres F to fill and press J to connect.

You can combine this procedure with Grid Fill - Ctrl + F > G

I don't know a better solution, I hope someone knows one because this is really good question.