Continuity mistake: After the main characters have been thrown out of the "lost city" by the rapidly rising waters, they are completely soaked (quite natural, as they were submerged in the well). But after everything dies down, all of the characters' hair and clothes are bone dry.
Continuity mistake: In a close-up, Indy removes the mask from one of the mummies and is holding it about a foot away from its face. In a shot from further away, Indy is holding the mask three or four feet away from the mummy's face.
Other mistake: In the warehouse scene when they're standing around the crate containing the heavily magnetized alien, all metal objects like guns and other items are pulled towards the magnetic force field. But Indy, who is standing next to the crate, has a bag with a metal strap that stays still when it also should've been pulled towards the alien.
Factual error: In the warehouse when Indy and the Soviets open the crate with the alien's body, all the overhead lights are pulled towards it. If the magnetic field was that strong, the lights also would have been pulled towards the box before it was opened. (Put a toy car on top of a table and pull it with a magnet from under the table to see the field in action).
Continuity mistake: In the shot where Mutt is on the front of the car sword fighting Spalko, when the shot changes he switches to the back of the car, still fighting her.
Continuity mistake: The drawing that Ox makes in front of Indy differs between shots: Watch the shape of the sun.
Continuity mistake: The Russians arrive at the Nevada base and salute the guards. The one on the far right is very close to the barrier, next to where the corner of the house is; half a second later he is several meters behind and walking towards the barrier again.
Other mistake: When Indy escapes on the rocket powered train, it is slowed down by a pool of water on the tracks. The only problem is that when this happens in real life, it's almost the same as slamming into a wall, and therefore Indy should have been flying several feet away from the rocket powered train. You can actually see the chair that the test persons before him used, and it has double seat belts on it.
Revealing mistake: When taking over the Nevada test site, the head Russian drops to tie his shoe as four soldiers pull out machine guns to kill the U.S. guards. Both soldiers on the right run out of ammo, yet keep moving their guns as though they are still shooting. (00:03:10)
Continuity mistake: In the beginning when the Russians are talking to Indy, his shirt is tucked, then in another shot it's untucked, then it's tucked again. This continues for about 3-5 shots.
Continuity mistake: Right at the start when the Russians pull Mac out of the car, they throw Indy's hat on the ground and there is a close-up shot of it where you can see Mac's feet right next to it. It then cuts to a relatively wide bird's-eye shot over the car as it is surrounded by Russian soldiers and there is nobody near the hat - Mac isn't even in the shot.
Revealing mistake: Right after the nuclear explosion and Indy comes out of the refrigerator, the scene changes to an airfield with two cars driving across the screen. The cars are obviously CGI, as the first one turns despite the wheels still pointing straight.
Continuity mistake: During the wedding scene, we see Mutt taking photos with a 1950s Kodak Brownie box camera. When wind blows Indy's fedora off the hat rack and up the aisle, Mutt stoops to pick up the hat, with the camera no longer in his hands. Indy snatches the hat away from Mutt as he passes, and the shot then cuts to a full view of all the wedding guests leaving the church, with Mutt in the center of the shot, still standing in the aisle the whole time. He takes out a comb and passes it through his hair, then picks up his jacket from the pew before he leaves, but the boxy Brownie camera is still nowhere to be seen. It's not in his pockets, and it's not in his jacket (the angular bulge would be more than noticeable).
Continuity mistake: Mutt points out at Orellana's mummy, whose shroud is tightened and neatly set. Indy walks towards it and the shot cuts to a closer angle where the shroud is now crappily set. Obviously there were previous shots of Indy opening it, and the shroud was later messily closed to repeat the shots.
Character mistake: In the scene with the ants, Jones refers to them as "Siafu", which is actually the Swahili name for driver ants which are native to Africa, not South America.
Audio problem: When Irina and Mutt are sword fighting in the jungle, the blade strikes all make echoing sounds.
Continuity mistake: When Indiana and Mutt are in the diner, notice the coffee cup Indy has on the table. In one shot, he has it in his hand. In another, his hand is not on it. This happens twice with the different camera angles.
Factual error: At the beginning of the movie where the convoy stops at the gate to the military base, the fence has "Razor Wire" on the top, not the period-correct barbed wire. Razor wire came into use in the late sixties by some industries, but the military did not use it until the eighties.
Continuity mistake: Before Indy finds the crystal skull, the mummy's head has a piece of cloth on its forehead. In a wide shot immediately afterwards, this cloth disappears.
Continuity mistake: After the nuclear explosion when Indiana Jones is thrown into the air inside the fridge, in one shot the doors are closed, then open, and then closed 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.