Cos the yanks were late as usual and therefore had more time to fit bigger guns...
Sorry.
.303 (7.62mm ish) was the standard British army rifle and machinegun round so I think that is why it was selected initially.
The standard fit for RAF single seat aircraft developed into 4 20mm cannon by the end of the battle of britain late 1940. This was based on the bitter experiance of german bombers wandering on despite being turned into sieves. It took some time to develop cannon for the wing mounts that were happy up to 30,000 feet and this was eventually solved with hot air feeds to prevent the breaches freezing.
So in fact they were more heavily armed than the Mustang (4 x 12.7mm) tho i suppose was really british as well.
It is debatable whether 4 x 20mm cannon firing an explosive shell was better than 8 x .5s(thunderbolt and others) firing non explosive shot.
Contempory german fighters, Bf109g (1943) had 1 30mm cannon in spinner and 2x 13mm machine guns. bf 110g (night fighter) 2x30mm 2x20mm rear gunner 2x7.9mm
This had evolved following experience starting in spain in 1936 when the 109 had just 2 x 7.9mm machineguns.
British bombers , Lanc 2 twin .303 and 1 quad .303,
American
B-17 9 x .5 (12.7mm)
The later british bombers designed for night use where the gunners were really more use as lookouts. Indeed there is some debate as to whether they would have been better leaving the guns and turrets behind entirely and getting better height , though Bomber command would probably have just put more bombs on.
The Americans had the knowlege of British use of the B-17 in daylight ops (disasterous) and decided to up gun it before they used it themselves.
So as you can see everyones ideas about aircraft armament developed through war experiance. Its just the Germans had a head start and the Americans gained a lot of British data.