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: During the beginning chase with Mutt and Indy, the bus goes from next to them in one shot to behind them in another.
Factual error: In the scene where they are flying to Peru from US, they fly over a "map" of a Central America region and the country of Belize is shown. That country was the British Honduras until 1973. It was then renamed to Belize.
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: 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.
Factual error: When Indy is getting decontaminated after the atomic explosion, the workers are using the M17 protective masks. These masks were not in use until the late 1960's.
Factual error: During the motorcycle chase seen (which was filmed in New Haven, CT) one can see modern yellow traffic lights covered in burlap in a few of the scenes.
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.
Factual error: During the fight in the rocket engine testing facility, the countdown timer has electronic digital numerals. In 1957, the countdown timer would have had flipping numbers, which are mechanically changed as the timer counts down, or a nixie tube display. See films like Fail Safe and Goldfinger for examples of such clocks or timers. LED's weren't even invented until 1962.
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: 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: 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 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.
Other mistake: When Mutt is riding his motorcycle the first time he meets Indy at the train station, his hand is pulling in the clutch as he rides. But with the sound and speed of the bike, the clutch would have to be out.
Factual error: In a shot of the exterior atomic test homes, we see child dummies playing on a Slip-n-Slide. That toy was not introduced by Wham-O until 1961.
Visible crew/equipment: As they are running away from the ants, you can see a rig of stage lights under the jeep Col. Spalko's crew was driving.
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.
Factual error: When Mutt throws the snake to Indiana Jones in the sand pit, it's an olive python (which he misidentifies as a rat snake). But olive pythons are not native to South America - they're Australian.
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.