In-battle fighter repair uses supplies. Crew use supply (Looks like 0.01 supply per crew/marine.) Over fleet limits use supply (not sure o fthe formula here though, but it's definitely not linear). Ship repair uses supply (over time during the reapirs). If you have fighters and a ship with a flight deck, the fighters get repaired immediate after combat using supplies (on top of any used during combat.)
The way I see it, your use of fighters is chewing through your supplies. I run a large fleet (100FP) of high-tech frigates with minimal crew and only use up 3 supply/day (I keep a stock of marines onboard). That's not including repairs post-battle. If you regularly use fighters, you're gonna need a decent cargo limit to feed them supplies during active usage.