This chart from a pilots manual ( from Zeno's Warbirds) pretty much explains why the P-47 was NOT considered for carrier use.
Even at 12,500lbs it needed 900ft of runway with a 40mph headwind for take-off and that is at 0 degrees C.and it had an approach speed of 115mph even at 10,800lbs (?).
F4U-1 Corsair had the following (also from Zeno's)
F4U-1 can take-off in 480ft with only a 30mph head wind while weighing 14,200ft. It's approach speed at 11,000lbs was 95mph.
And no, catapults are not the answer. The US Navy wants to fly as many planes as possible off the flight deck in the least amount of time. Only enough planes are catapulted until the remainder have enough room to take-off with out catapult.
Flying them off "light" (little or no ammo, only enough fuel to make shore base, perhaps not even full number of guns (or none)) in no way means they could undertake operational sorties on a regular basis. The different approach speeds, in addition to making things more difficult, also means the the arresting gear has to dissipate about 46% more energy even if the planes weighed the same.
The P-47 may have been rugged but it would be interesting to see what would have been needed to make it stand up to 60-100 arresting hook landings.