I support
@Wraith11B and
@TheButterflyComposer 's assessments. As long as you stay out of British-claimed territory, anything Axis or Vichy is fair game. Let the British keep Egypt and Israel-Palestina, but take Syria, Lebanon, Libya, Algeria, Tunisia, Morocco, Sardinia, even Corsica could be fair game, if only as a bargaining chip. The US definitely won't care who holds those territories, as long as they remain stable and peaceful. They'll be happy for you to use those USMC Divisions in that area, if only for the propaganda value of it.
Fast Divisions don't need to be very powerful in North-Africa, you can probably get away with binary Motx2 or L Arm, Mot Divisions (maybe with some armoured cars sprinkled in for some Combined Arms bonus). If you can cut of the enemy's supply line, the rest doesn't really matter that much. There is definitely the potential to grab a lot of territory for only a little investment. Any Mediterranean real estate has strategic value, especially if it has a sizeable port and/or Air Base on it, not to mention the resources that can be found in the area. If you manage to take all that before going into Italy proper, the Med will almost become a Turkish lake, with only the côte d'Azur, the Spanish Coast, Egypt, Cyprus, and Malta in foreign hands. This is the best possible scenario, but even getting half of all that will be an impressive achievement.
Once the Axis is beaten, you can always go for round 2, and take over the remaining British territories, and maybe even the south of France. Considering the US is in the Comintern, they'll rather see those territories in the hands of the Turkey, than in pesky British hands. Then, once Comintern dominance is established, you can even go for Iberia, and possibly even India, though that's a stretch. As long as you keep the Americans happy, and don't start messing with the Soviet sphere of influence. There is an added benefit to all this. The more territory it grabs, the more Turkey becomes a real Major power, and an equal partner, in a post-war trilateral USA-USSR-UGNR world order.