OpenGL 3.2 Core Profile glLineWidth
Using OpenGL 3.2 core profile, calling glLineWidth with a value greater than 1.0 give an INVALID_VALUE error (call glGetError to prove it).
Surely you can get the wanted result by determining the quad required for drawing the line.
I think you should be able to generate quads from from line points: hey, a larger line is a quad! Maybe you could use techinques like this to get your wanted result.
The key is: instead of rely on LineWidth, you give a unit quad as input (4 vertices using triangle strip), then transform incoming vertices inside a shader passing to it appropriate uniforms.
Maybe another approach would be rendering using a geometry shader: generate a quad from a point. However, I'm not sure about this point. I don't know if a geometry shader (only if it feasible, of course) would be the best approach: the cost of drawing a line strip using a single quad would be the shader uniform setup for each line composing the strip.