Continuity mistake: In Akator, Indy, surrounded by Marion and Mac, tells Mutt to find a stone. When Mutt comes back, in less than a second, Marion and Mac are nowhere to be seen.
Continuity mistake: When Ox finds out about the stone that pours out sand, Mutt is up the stairs. Then the angle changes and he is down, starting to walk upwards.
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: When Jones is strapped to the chair in the tent Irina uncovers an alien to shows him. She then covers the alien with the cloth, but when the camera moves to a wide shot the soldier at the other side of the table does the same thing.
Continuity mistake: At the diner, Mutt places the photo on his hat, yet it swaps from standing out to tucked inside, between shots.
Factual error: After Indy finds the skull, the map to show his travel has all the words written in Spanish, with the accents properly placed, except for the words 'Cuzco' and 'Río Amazonas', misspelled in English, 'Cusco' and 'Amazon'.
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 Col. Spalko begins cutting the leather covering the alien corpse in the casket, no straps are seen. In the close-up shot a strap is suddenly visible.
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: 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.
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.