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: When Indy exits the fridge after the blast, he leans on both of his arms and faces a gopher. From a different angle, he is leaning on his left arm only.
Continuity mistake: Indy empties the fridge hurriedly and throws away all of the trays, then pulls away the last drawer below. A shot later from a different angle, he is throwing away the penultimate tray and pulling out the drawer again.
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: 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: 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: 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: 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.
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.