Continuity mistake: When Spalko climbs a hanging vine to escape the large, marauding ants, one jumps on her leg. As it crawls up, she smashes the squishy bug between her knees (you can hear it splatting), but there's no stain or body parts left on her very clean pants.
Continuity mistake: In the beginning when Indy's hat is thrown from the trunk onto the ground, it faces the 9 o'clock postion. But when Indy reaches to pick it up half-a-second later, it faces the 12 o'clock direction.
Continuity mistake: When Indy is held captive in the Soviet's tent, Mac enters and they have a conversation. Indy then turns his head to his left, looking straight at Mac. Just before Indy makes his threat about what he'll do to Mac when he gets loose, there is an angle change, and Indy is suddenly staring straight ahead, with no time to turn his head in between shots. He then turns his head back to look Mac in the face again.
Continuity mistake: Throughout the whole movie, Irina's hair seems to be kept absolutely pristine, even when she is flung from cars. This is very noticible at the car chase in the military base when she is thrown from the car by Indy. One minute, a few strands of hair are slightly messed, then they are fixed again. It is likely, however, given the many other nods to films of that period, that the "concrete hair" effect is a deliberate decision.
Continuity mistake: In the diner scene between "Indy" and "Mutt," Indy's shirt-collar points are smooth under his jacket lapel in some close-up shots, but in others one collar point is curled up.
Continuity mistake: When Mutt first shows up, he is riding his bike between two train wagons, or between a train and a railing next to the station. This changes back and forth with no logic at all.
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 siafu ant attack, while fisticuffs ensue between Indiana and Dovchenko, note the direction of Dovchenko's belt, with its buckle prongs to his left. Then, when Dovchenko falls backwards and the ants swarm all over him, his belt is worn around his waist the opposite way, with its prongs to his right (neither are flipped shots). (01:26:55)
Continuity mistake: When Indiana is with Mutt while the boy rides the motorcycle, Indiana's shoes change; note the soles change from heels to flat rubber-type soles, then back to heeled soles.
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: 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: 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: 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 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: 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.
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 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.
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.