I am making a 3D logo.
When I create a text object and use solidify mod to make it thicker it does this:
When I don't use the solid mod, and instead convert it to a curve to do a curve extrusion it does this:
Why is it doing this?
The missing fill in your curves is caused by overlap. If the curves self intersect Blender doesn't know how to fill them correctly.
Again if you are working with curves don't use the solidify modifier, instead use the curve extrude option which should yield better results.
That being said, I think you started on the wrong foot, if you plan on curving your object the way you show in the image you will never get a clean solution from curves. Because of the triangle based topology of curve object you will never get a clean smooth curvature.
You can try and minimize it by using a Subsurf modifier but it will generally maintain it's triangulated appearance.
You should do this from scratch with a mesh object with an even quad based topology