Well, for example, here's the polygon I want to create a normal of:
Without waving my hands around trying to visualize those vertices
, I'll take a guess that the points don't lie in any common plane - this is generally a bad thing! Non-simple/self-intersecting polygons, non-convex polygons and polygons with holes in are not guaranteed to be rendered correctly. If the points of a polygon don't exist all on a single plane, it's quite easy for that polygon to be self-intersecting after viewing or modelling transformations.
Either break that polygon into smaller polygons, or (better) draw the surface as a triangle strip.
And my graphics card is an S3 Savage IX/MX with 8MB.
Sucks, don't it?