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: 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: 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: 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: 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: 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: In the wedding scene the bride has open toed shoes when she kisses Indy and begins to exit the pulpit area, but different closed toe shoes as she and Indy depart the church.
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 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: 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.
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.
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.