Continuity mistake: While Indy and Mutt are walking by the market stalls, there's no continuity with the people behind them. Sometimes even a group of men disappear only to reappear several shots later.
Continuity mistake: During the fight in the bar, when Indy and Mutt exit, a man to the right hugs a woman in green to protect her. A frame later they're 3 meters apart.
Continuity mistake: In Nazca, the two locals Indy talks to about Oxley teleport next to the bus door between shots and the large wagon behind Indy disappears. (00:39:28)
Continuity mistake: Dovchenko smiles and raises his pistol. In the next shot, he repeats the action. (00:13:34)
Continuity mistake: In Perú when Mutt is sweeping the floor Indy holds the map with one hand. A shot later he holds it with both.
Continuity mistake: As the Dean enters, several pictures hang on the wall. When they leave the pictures have been changed. (00:24:40 - 00:25:45)
Continuity mistake: The gold mask on Orellana's face changes angle between shots.
Continuity mistake: Spalko steps on a tree to free herself from the ants, 2 meters away from where a soldier is starting to be eaten. A frame later the soldier is gone, there's not even a small mound where his corpse could be.
Continuity mistake: When the group enters the final temple the skull falls into the water. Mutt picks it up, and there's a shot of Mac grabbing some gold. Then back to a wide angle and Mac is repeating the previous movement with a totally different hairstyle.
Continuity mistake: After Mutt points at the waterfall, Indy grabs the skull and Mutt leans on him to raise up, yet a frame later from a different angle his hands are away from Indy.
Continuity mistake: When the raft is about to fall through the second waterfall, there's a huge rock to the left which disappears in the wide angle.
Continuity mistake: When Indy removes Orellana's golden mask, the light changes abruptly from very shiny to obscure and mysterious.
Continuity mistake: During the ant scene, the branch that Indy uses to knock out the Russian wasn't there in the first place, it magically appears between shots.
Continuity mistake: After Mutt's bike knocks down the "Better red than dead" banner it falls sideways; yet when the chasing car arrives, the banner is magically up again, ready to fall horizontally on the car's windshield.
Audio problem: During the chase inside the campus, a car destroys some stone vases and a blonde woman cries in terror while running away, yet her mouth is not moving.
Continuity mistake: When Indy and Mutt's bike is about to exit the campus, a green car is passing by, which disappears in the immediate wide angle of the street.
Continuity mistake: When Mutt and Indy ride inside the arched door in the campus, a couple runs away, yet appears still there from the opposite angle.
Continuity mistake: When Indy and Mutt are about to enter the university, the guard is holding both gate doors almost closed. A frame later he's holding one completely opened.
Continuity mistake: When the Akator temple starts to move, the sand by the groups' feet keeps changing from sandy with a couple of rocks, to filled with weeds and leaves, and back to sandy.
Continuity mistake: Inside the waterfall temple, the group walks by a room (with a passage on each side and a double entrance with steps in the middle) where natives appear and columns and walls fall off. The angle changes to a seemingly new room, but it's obviously the same place, with the camera angle slightly turned to the right. All the previous people are magically gone, the debris has disappeared, and natives are coming out of different places. Really odd editing. (01:30:35)
Answer: People felt it was ridiculous and cartoonish, even by Indiana Jones standards. Even if it was possible to survive a nuclear blast via the lead lining of a fridge (it's not), or that the fridge would simply be thrown away rather than be melted/torn apart like everything else in the vicinity (it wouldn't), the impact of being flung what appears to be a mile or so through the air, then violently crashing into and rolling over the ground, would certainly kill anything inside. The controversy arose because usually, in "classic" Indy films, the fantastical elements were exactly that: fantastic, magical, and/or supernatural. This was presented as taking place in our reality, with no "power of God" or magic spells, and for many, that was just too much disbelief to suspend.