Had the same issue with a WordPress site I built 4 years ago. Monthly bandwidth kept creeping up every month to over 10G. Scanned for malware, disabled trackbacks, cleaned out old posts, and optimized database but nothing worked.
Saw somewhere that it could be the theme, so I switched to Twenty Fourteen and adjusted colors, etc to look like old theme. That was two weeks ago and daily bandwidth is now 0.4% of what it was before.