I am trying to export a simple gltf model to test how bump mapping looks in my engine. I followed this link but nothing I do seems to be working. The exported model always has dark spots and looks pretty bad. What exactly is the process to export normal map from blender to gltf?
You're using the wrong node "Bump" instead of "Normal Map" for glTF export. The glTF format only accepts a Tangent-Space Normal Map, not general bump maps, for rendering efficiency. If your image is greyscale (a heightmap), you'll need to convert it to a normal map as well. The color space of that image should be set to non-color.
This is described in more detail in the Blender Manual's glTF Export section.