A little statistical comparison of RAF and USAAF losses.
From what resources I can find online (usual caveats as to reliability apply) Bomber Command lost, across all theatres and all aircraft categories, 12,330 aircraft.
This includes all combat losses, as well as training and accident losses of bomber aircraft, as well as aircraft written off for other reasons, such as being sunk while being transported in convoys to Russia.
I'm unsure if this includes losses of trainers, liaison aircraft and the like, but I doubt it. I'm sure more aircraft were written off in training before pilots were assigned to actual units.
In the ETO, around 8,950 Bomber Command aircraft were shot down/failed to return, crashed on take/off landing, were impounded German/foreign countries or written off from damage from ops.
In comparison, USAAF bomber losses in the ETO in combat were 5,548 heavy bombers and another 815 medium/light bombers, for a total of 6,363. This includes accidents. First loss was August 1942.
So we con conclude that the majority of bombers shot down in the ETO were from Bomber Command. In Europe, Bomber Command lost about seven bomber aircraft for every five bombers lost by the USAAF.
However, given that USAAF ETO bomber ops did not begin until around August 1942 and didn't really get into full steam until mid 1943, the number of losses from mid-1943 onwards were in all probability about the same for Bomber Command and the USAAF.
There were 121,867 USAAF personnel casualties during WW2, of which 94,565 were in theatres against Germany and 63,410 in the ETO. So, in the air the majority of casualties were in battles in the ETO. There is no break-down between fighter and bomber crew casualties though.
Expanding beyond the ETO, USAAF bomber combat losses in all other theatres were 5,687 very heavy/heavy/medium/light bombers. This gives a total of 12,050 USAAF bombers that were combat losses.
However, there were another 479 bombers were lost en route to various theatres.
So, there were just over 12,500 USAAF bombers lost outside the continental US.
Total USAAF losses in all overseas theatres were 41,575 combat and trainer/liaison aircraft. This means that bombers accounted for 30% of all USAAF losses.