Aha, you've discovered Thing One about photography, which is: daylight is bright. Very bright. So bright you'd need a much bigger light than that to avoid your head being outlined by bleached-out, overexposed whiteness.
Considered closing the curtains? The next thing to do is to bounce the light off a nearby white surface, such as a wall, or if you're feeling flush, a chunk of white polystyrene insulating board. The result will be a light source of large effective size, which will produce softer shadows and avoid throwing a big shadow of your nose across your face.
A traditional lighting setup for the sort of material you're doing (which we'd generally refer to as a "piece to camera") is three point, in which case you'd have a key light about 45 degrees off the angle between you and the camera - which is sort of what you're doing with that desk lamp. Sometimes you then need a fill light on the other side of your face, to fill in that dark shadow that your webcam can't deal with, so we can see your other eye. And then, because video is generally two-dimensional, a backlight behind and above you, for that shiny-haired newsreader look.
You can absolutely implement all of this with stuff you can get at B&Q or on ebay for cheap. Think about using low-energy fluorescent lightbulbs, if not because they're green, because they won't cause you to melt.