Continuity mistake: In the alien room, Spalko walks to Ox to take the skull away from him. In the first angle Marion is standing to the left of Ox, but in the immediate close-up she is behind him.
Continuity mistake: After Indy has been decontaminated, the lamp behind him swaps from lit to unlit between shots.
Continuity mistake: When Indy and Mutt are in the diner, two baddies show up, Mutt pulls out his knife and places it leaning to the left. In the close-up it's facing upwards, despite Mutt not having moved it.
Continuity mistake: In the Nevada warehouse,when Spalko is opening the corpse bag, the amount of smoke coming out swaps from a lot to zero depending on the angle, without any logical sense whatsoever.
Continuity mistake: In the Nevada warehouse, the first angle of the corpse's hand shows the baby, ring and middle finger together. When the angle changes, the position of the fingers is completely different, with the baby finger now totally separated from the rest.
Continuity mistake: In the graveyard, Indy hits a native with a shovel and knocks his mask and wig off, revealing a short thin hair. A frame later, when the native escapes, his hair is back to fuzzy and long.
Continuity mistake: In the Nevada warehouse, when Spalko opens the corpse's body, its hand is covered in a white dust. A frame later it's clean.
Continuity mistake: In the Nevada warehouse, the way the whip is rolled around the soldier's shoulder keeps changing between angles in a matter of nanoseconds.
Continuity mistake: Indy enters the warehouse and two soldiers on the right gather next to him, while Mac walks towards Indy. When the angle changes, the guards are standing several meters behind, walking towards Indy again.
Continuity mistake: The military car racing against the young kids has its right windshield covered in dust, except for the wiper's track. A frame later the glass is clean.
Continuity mistake: While Dovchenko is approaching Hangar 51's gate, there is a guard standing behind the STOP white sign written on the barrier. In the next shot, that guard has teleported next to the fuse box. (00:03:03)
Continuity mistake: Indy asks for shotgun shells and a soldier drops some inside Dovchenko's cap (which is in Indy's left hand). In the next shot, they are suddenly in the same hand that held the cap, which is now in Indy's right hand. (00:10:00)
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.
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 Mutt and Indy run away on the bike, there's a shiny red car parked behind. A mere frame later, a dusty greyish car has magically appeared parked next to it.
Continuity mistake: When the race begins between the staff car and the hot rod, they immediately pull away from the convoy, as is logical. However, in the next shot the convoy is right behind the staff car again - which is still accelerating racing the hot rod. When the convoy pulls off the main road towards Area 51, the convoy again is right behind the staff car. (00:01:40)
Continuity mistake: At the start, after Irina Spalko unsuccessfully reads Indy's mind she slaps him a few times. Then she lowers her hand, but in the following close-up the hand is raised and being lowered again.
Continuity mistake: After Mutt crashes his bike in the library there's a shot on a nerd. Watch the table behind him. A lamp suddenly appears in the second angle.
Continuity mistake: During the credits, while the young kids are racing the military car, the jeep behind always has a roof, as seen in all the outside and inside angles. However, in barely half a second, when the cars turn towards the base, the jeep has no roof.
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.
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.