If you have the mesh as you show it, then you should be able to select it and go into uv face select - make sure you select all the faces, then hit ‘u’ and then choose ‘project from view’ and you’ll see the mesh lined up in the uv image editor. you can then select all the uv’s there and scale up/rotate,etc to line it up to your texture image. Be sure to load up the same image in your material’s texture channels, and set it to ‘uv’ instead of ‘orco’. The triangles shown are because the curve is filling the easiest way possible for the profile you’ve made at the resolution you’ve set - a work around would be to subdivide a plane many times and proportional edit tool it to get it to fit the same shape, but with quads. I’m sure there are other techniques you might find as well.