Were WW2 American fighters designed to operate from dirt airstrips?
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Opening theaters in the Pacific, for example, using fighters/bombers fitted with bush tires would not have been a good choice indeed. Maybe one pilot can pull it off. But not all, and certainly not the damaged planes.

As planes got heavier before WWII, and especially during WWII due to the massive engines and the all-metal construction, came the need for inflated tires, paved runways, hydraulic brakes...

... and the need for the Marston Mat. By using perforated steel planks, a runway can be built almost anywhere in a relatively short time (source).

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Above shows a North American P-51 taking off from Iwo Jima. Note the pattern on the ground. From far away in other B&amp;W images the pattern does not become clear and the planes look as if they're using pure dirt runways.


  Marston Mats were among the most important inventions of World War II (airspacemag.com).


As seen in your other question, How are modern jets modified to takeoff/land on a dirt runway?, Russian fighters are built with unpaved runways in mind. As are modern military transports. Transports have the luxury of space for many tires. But for jet fighters, it was the advances in tire manufacturing, which itself was due to the jet age that brought the swept wings and consequently the faster takeoff and landing speeds.

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Of course a humid tropical rainy island is different from a bone dry desert. The different soils and the water content affect their bearing and shear strengths. In North Africa, you may not need special reinforcements for the soil where its strength can be 4 times as high (think dry lakebeds).