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 aliens kill the Russians, the group is launched out of the temple thanks to a jet of water. Seconds later they're all completely dry.
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: By the waterfall Indy says "Nobody else can" and then Mutt's wet hair swaps from brushed backwards to messy.
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.
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.