1. He says 150 light years out of 100,000 light years. Then he writes 50/100000, just losing a factor of 3 for no reason at all.

2. A typical sheet of writing paper is 10um or so in thickness and 20cm or so in extent. The ratio is 1/20000, which is 10x smaller number than even his (wrong) 50/100000.

3. His figure of 150ly is based on the distribution of molecular clouds. This seems an absolutely ridiculous way of measuring the thickness of the galaxy, about as sensible as measuring the dimensions of a human being by the distribution of their stem cells or something. It seems that the choice was motivated by wanting the smallest possible thickness figure.

4. The figure of 150ly for molecular cloud heights is too small. See figure 3 of http://iopscience.iop.org/1538-4357/619/2/L159/pdf/18343.web...; the right-hand portion of the figure shows average heights, which indeed are in that range, but as you can see from the left-hand portion there's a lot of variation and there are plenty of molecular clouds well above 150ly. (Note that the vertical scale in that figure is in parsecs; a parsec is about 3.26ly.)

So. The author of that page (1) selected the "thinnest" measure he could find, (2) used too low a value for it, (3) dropped a factor of 3, and (4) still got a result 10x thicker than a typical piece of paper.

(That's assuming that "flat" means "thin". The Milky Way is not particularly close to being planar, either.)

Wikipedia says the stellar disc of the Milky Way is 1,000 light-years thick and cites http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/980317b.... . The core is thicker. Even the image on the page you linked to shows an object that is much thicker than a piece of paper.