As above, to test for colourless odourless and poisonous carbon monoxide.
Other products of combustion are colourless and odourless carbon dioxide and water vapour, plus traces of sulphur oxides. It is perfectly possible that the dizziness could come from an excess of carbon DIOXIDE in the air around the outside of the bathroom window, being drawn into the room because of the extractor fan. This is not poisonous, but is no good for breathing either. Have you tried sealing the window up to see if this changes things?