The OP said that the Germans managed to get the Turks on their side or at least be a non-belligerent willing to grant military access . I agree with you that, the later on this is, the less likely the Turks are to buckle, meaning the intimidation would have likely had to have happened in 41 or 42. However, the simple fact is that with Barbarossa Hitler has whole army groups to play around with (Some need to garrison the Eastern Front, granted... but there they can take advantage of the strategic defensive. Since the crux of so many of Germany's problems were logistics-based, being able to operate defensively from pre-built airfields, use your own pre-built rail lines for transport, and having facilities set up to repair/manufacture parts for damaged aircraft and vechiles will be a huge force multiplier). Turkey's position really isen't enviable: they're getting squeezed from three sides and each one has the capablity of launching a crippling strike before any potential allies can get in place to protect them.
Any way you slice it, though, the land campaign is going to be a nightmare. I expect to see the Brits and Americans argueing endlessly as the British aren't going to be willing to spill alot of their (very limited) blood and are going to be stuck constantly looking for weak points that don't exists.
Turkey becoming an Axis member or a Sweden-like neutral that allows Axis troops in is going to be the alarm that awakens Stalin. Have a look at the German-Soviet diplomatic correspondence running up to mid-1941: much of it is about containing German meddling in the Balkans and then Turkey.
With German troops all along his Western border and in Turkey, Stalin not only stops the supplies, but he'll also not be deluded into thinking the Germans won't attack. Additionally, he might very well be the one attacking, if and when the Westerners try something significant in Europe.
As to weak points, the Germans, as long as they haven't fielded the Luftwaffe in the East, can easily defend the most obvious one, Ploesti and the Danube. But they're still stuck with a continent that is not self-sufficient as to food. Once they lose North Africa (and they will, even if they have a surplus of troops - that doesn't mean they have a surplus of shipping and port handling capacity in the Med), things get worse; that's because, as long as Vichy France was a nominal neutral, the British allowed grains, oil and fish from the French North African colonies to go to Marseilles, but when that ended, those supplies ended too. The Germans aren't going to draw food from the Ukraine either. If the British aren't making the Archangelsk run effort, they can devote Royal Navy resources to the Med and, once North Africa is cleared, to put pressure not only on Italy (obviously) but also on neutral but unfriendly Spain and Turkey. The pressure needn't be overtly military, it can come with a carrot: sell your strategic stuff to us, or else your cargo shipping will suffer from delays as our anti-contraband measures get tighter. And Spain, needless to say, was importing food from South America. The British can also reinforce Cyprus and Syria, just to let Turkey know they shouldn't exaggerate. Istanbul can't withstand a bombing by the Luftwaffe - but neither a series of night visits by British bombers.