Continuity mistake: During the jungle chase, while Spalko fires at Indy's jeep, her hair remains perfectly still in the angle shot from behind, but moves wildly in the front shots. This swaps back and forth.
Continuity mistake: During the ants scene, Indy is about to grab a stick to the left of his hat. In the close-up the stick and the hat have swapped sides.
Continuity mistake: In the Nevada warehouse, while Spalko opens the bag a strap next to it appears or disappears depending on the angle.
Continuity mistake: When Indy is about to grab the branch to hit the soldier, his hat keeps moving around between shots.
Continuity mistake: When Mutt meets Indy, Mutt has a train behind, yet the reflection of the glasses from Indy's wagon don't reflect it, instead showing a sky and trees.
Continuity mistake: After Mutt falls off the graveyard's ladder and is attacked by a native, the scattered broken parts of the ladder disappear to allow for the fight scene to take place.
Continuity mistake: When Indy and Mac are aiming at the soldiers in Nevada the distance to the Jeep behind increases noticeably between shots.
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.
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.