Continuity mistake: At the beginning when Indy is talking to Spalko, his hands keep alternating from being in his pockets to just resting at his sides between shots.
Continuity mistake: After Indiana and team fall off the disappearing stairs as they make their way into the lost city, they are walking around in the water. But when they step through the next door into the treasure room, all are completely dry.
Continuity mistake: When Indiana and Mutt are in the diner, at the start of the scene there is a pretty blonde girl sitting at the counter just past the corner. A moment later, her seat and the one next to it are occupied by the two Russian guys in suits, and she is talking to a guy with glasses who was earlier by himself. Not enough time had passed for that to change.
Continuity mistake: When Indiana Jones is locked in the chair at the Russian camp, there is a reel-to-reel tape recorder machine running to Indy's left. When you first see it, the amount of tape on the left reel is low. Then, when Irina Spalko walks in, she goes over to it and flicks a switch. As the camera moves out, the tape on the left-hand reel is suddenly nearly equal to the one on the right.
Continuity mistake: In the tent, Indy's wrists are tied and he's facing the crystal skull. From one angle his hands are clenched fists, but from a different one they hang loose. The difference is noticable because the angles keep swapping and the hand positions are not continuous.
Continuity mistake: In the forest car chase, towards the end of Mutt's sword-fight, we see Indy driving a car behind Mutt. It cuts to Mutt picking up the skull and then back and Indy's car has disappeared. Then, as Mutt falls off the car, magically Indy appears in time to catch him.
Continuity mistake: When Indy and Irina are standing in the foreground analyzing Ox's drawings, look over Indy's shoulder and you will see Ox, Marion and Mutt standing up and moving forward. After the scene cuts a couple of times, a soldier asks them to stand up and move forward again.
Continuity mistake: During the beginning chase with Mutt and Indy, the bus goes from next to them in one shot to behind them in another.
Continuity mistake: When Indy and Mutt first meet in the diner scene, look out the window behind Indy. A group of leather-clad bikers are sitting and talking on their motorcycles. They vanish and reappear at least three times during the scene.
Continuity mistake: In the jungle chase scene, Indy and the others blow up the front-running plough machine. They then continue to drive around in a ploughed jungle alley. They should be hitting a lot of trees.
Continuity mistake: Just before the Russian backhands Indy in front of the warehouse at the beginning of the film, his hands are crossed behind his own back, but immediately in the next shot, they're crossed in front of him.
Continuity mistake: After visiting Ox's jail, Indy arrives at the ruins and it's pitch dark. He enters the burial room several minutes later and sun beams are now coming through the ceiling.
Continuity mistake: When Mutt slams his fist on the table in the diner, the ketchup and mustard bottles fall over. When the shot pans to Indiana's back, the bottles are standing up again. He then tells Mutt to sit down and the camera pans back to Mutt. The bottles are lying down again and Indiana picks them both up.
Continuity mistake: When the gang arrive at Akator, and Indy pulls the first rock from the wall and sand starts pouring out, Mutt runs out from that shot to find himself a rock but in the next shot, when Indy says to Mac, "Go get your own!(rock)", Mutt is suddenly back in the shot and runs off to find a rock, again.
Continuity mistake: In the opening scene with the drag race, the lead car in the caravan speeds ahead in response to the challenge. In the next scene, the caravan turns right and the challenger continues straight. At this point, the entire caravan is together despite the fact that the lead car would have been far ahead by that point.
Continuity mistake: When they fall into the water, while the stairs disappear, the skull splashes twice.
Continuity mistake: When the civilian car is chasing the military convoy approaching the Nevada test site it crosses the convoy between the third and the second truck. It then passes the next two trucks on the left. In the next shot it is again behind the first truck. (00:00:55)
Continuity mistake: After the duel between Mutt and Irina Spalko, Mutt falls down into the car with Indy and yells, "She's getting away!" as she is seen driving off. However, in the following shot when they are driving near the edge, Irina comes from behind them.
Continuity mistake: When Mutt and the Russian woman are fighting in the normal view, you see the the AK-47, but when it changes to overhead, it disappears.
Continuity mistake: When Indy and Mutt meet in the diner, the spoon on the table changes its position between shots.
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.