Considering it is 1944, it is probably a bit late with the advice, so I will talk in general about how to eliminate the Luftwaffe.
In the beginning.....you should only be bombing the Ruhr with Bomber Command. At Night. Set up a good bombing Air Directive, say Wuppertal with a 3 hex box. Don't bomb every day, and every turn, rest any squadron with a morale less then say 60.
You can rebase all the night fighters to southern England and make them part of Bomber Command if you want, but sometimes you may want to use some of them for other things.
The americans should also start by bombing the Ruhr....but can occasionally range elsewhere. As you get more and more long range fighters, you can hit deeper targets.
If for any reason you suspect the Luftwaffe has a presence in airbases in any area...bomb the airfields with the American 4 engine bombers. Carpet bombing the airfields will wreck them.
Feel free to fly all the air superiority missions you want....trading fighters for fighters only favours you. If you are resting a lot of bomber squadrons, you can keep the attrition up.
Finally..at some point the back of the day Luftwaffe fighters will be broken....you no longer have to fly at night with Bomber command, so switch them to day missions and bombs away.
“My logisticians are a humorless lot … they know if my campaign fails, they are the first ones I will slay.” – Alexander the Great