Continuity mistake: When Indy and Dovchenko are shot out of Hangar 51, the expression on the Russian's face changes between shots. (00:17:00)
Continuity mistake: After Mac betrays Indy again, and Indy asks him if he's a triple agent, Dimitri suddenly appears next to Mac before Spalko enters the room.
Continuity mistake: While the soldiers are loading the coffin inside the jeep, you can see Spalko on the left side of the jeep, handing his rapier to Dovchenko. In the next shot, she is next to the back of the jeep with the rapier again and repeats the handing action. (00:11:39)
Continuity mistake: The Russian convoy consists of one staff car, two jeeps, and three trucks. When the convoy arrives at Hangar 51, a second staff car has appeared, from which Irina Spalko steps out. (00:02:44 - 00:05:50)
Continuity mistake: While the staff car and the black car are racing, there's a wide shot of the place, with only the road and some mountains in the background. In a few shots later, the Atomic Cafe has appeared from nowhere. (00:01:52 - 00:02:20)
Continuity mistake: When Indy asks for shotgun shells, behind him are, from left to right, Ivan, Spalko, Dovchenko, and Mac. Just behind Spalko, stands Roosevelt. But, when Indy asks for pliers, Roosevelt has appeared next to Indy, Spalko has moved to the right side of the group, and Ivan has disappeared. (00:09:57)
Continuity mistake: When Indy and Mutt exit the bar, there's a man and a young girl close to each other, plus a man in brown 4 metres behind, running towards them. A shot later, the man and girl are a metre apart, and the man in brown is right next to the young girl. Also, the man in a gabardine that walks by when Mutt gets on his motorbike has no continuity between shots.
Continuity mistake: When Indy tells the group to run away from the ants by jumping off the truck, Mutt jumps out, followed by Oxley. Shot changes, and Mutt is back in the truck, and Oxley is already out.
Continuity mistake: After they fall inside the temple, Mac steals gold from the skeleton and turns around. Shot changes, and he is turning his back on us, repeating all previous movements.
Continuity mistake: A car ploughs through huge iron gates, ripping them from the their hinges. Seconds later the car is pristine.
Continuity mistake: When Indy and Mutt are seated in the diner, Mutt stands up fast in an argument between the two, causing the bottles of ketchup and mustered to be knocked over. The shot then alternates from different directions, making the bottles appear as both standing and knocked over. When Mutt finally sits back down Indy sets the bottles back upright.
Continuity mistake: After the monkey chase scene, after Spalko's jeep lands on top of Indy's, Indy, Mutt, Mac and Oxley start to raise, but when the angle changes they're crouched again, plus in a completely different position.
Continuity mistake: Oxley takes the skull out of the bag and places it behind his hand to scare the ants away. When the angle changes there's a brief glance of his right hand and the bag is gone.
Continuity mistake: During the quicksand scene, when Mutt is throwing a snake at Indy, Indy's left (our right) sleeve swaps from covered in sand to spotless between shots.
Continuity mistake: When Mutt and Indy enter the library, a chubby man in a suit holding a brown briefcase runs away, yet appears calmly walking half a second later from a different angle.
Continuity mistake: Mutt's bike passes by a group of strikers in the campus, and heads for the red banner. When the angle changes, a second group of strikers has suddenly appeared out of nowhere next to the banner.
Continuity mistake: When Indy is about to hop on Mutt's bike, a man walks behind him twice. First time slowly, second time fast.
Continuity mistake: At the Mexican market, while Mtt is playing with his pocket knife, watch the chrome part of the light: it reflects a barren place with a brown squared building. When the angle changes it all changes to a bus and a lot of people hanging around.
Continuity mistake: Inside the waterfall, Ox takes out the skull, moves his hand from the top to the side of the skull. The angle changes and the hand remains on the side, but when the angle changes to behind Ox his hand is on top of the skull, repeating the first movements.
Continuity mistake: Inside the waterfall Ox takes out the skull while Marion watches from behind. When the angle changes, her hand is suddenly on his shoulder.
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.