I tried a tons of times, but the key is the delay turn. If you put our fighters in Air Superiority Mision this turn, it will make effect the next enemy turn, not the inmediatly next, the another one.
Yes, the Air Superiority (AS) mission is bugged in that it always activates 1 turn 'late' -- even if you give (Jet) Fighters M=2.
@Kirejara investigated this awhile ago; you should be able to locate that thread via his user-page.
(And yes, Jet Fighters + F-15s
can also do AS-missions -- it's Stealth Fighters that can't)
Air-unit bombardment of cities will also always hit air-units first, before it hits anything else (just like sea-unit bombardment always hits sea-units first). So you either need to protect your cities with AA-units (or later, a SAM Battery) as
@vmxa has noted, or rebase newly-built air units to an Airfield, where they can be protected by fortified land-units (in the sense that the land-unit will get bombed first).
(If you build an Airfield outside your city, and set it as the Rally Point for that city, this re-basing will happen automatically
during the interturn when the unit is built, so you can then immediately start assigning the unit to 'proper' air-missions).
It might also be a good idea to turn off "Cancel Orders for Enemy Combat Unit" in the Preferences (Ctrl-P) menu, to prevent your AS-missions from being turned off by any enemy attack.