Continuity mistake: Mutt says, "We have to cross the waterfall," and extends her left hand. From the opposite angle, a frame later, he is extending the right arm.
Continuity mistake: When the Russians arrive at the barrier the sky is either sun setting, bright and sunny, or cloudy, depending on the angle.
Continuity mistake: When Indy finds out that Mac has betrayed him, Mac is standing half a meter away. Right after, from a different angle, he is 10 centimeters away, with the tip of his rifle touching Indy. And right away, from yet another angle, the gun is on top of Indy's shoulder.
Continuity mistake: When Mutt is pulling Indy out of the quicksand both are inches away from each other. A frame later Mutt is meters away in a different position, as if he had fallen backwards.
Continuity mistake: Indy's car crashes face to face against Mac's, destroying its front. When Indy runs away there's a view of the car and the damages are minor.
Continuity mistake: After Mutt falls from the ladder in the graveyard, a native jumps from above to attack him. Indy is already on the right side of the scene, walking out, but when the angle changes he is standing on the left side, very close to Mutt.
Continuity mistake: When Mutt's ladder to the temple ruins breaks and he falls to the ground, the position of the broken parts around him keeps changing positions between shots.
Continuity mistake: In the warehouse, after Indy throws Spalko out of the car he is about to crash against Mac, who grabs the car's window totally panicked. A frame later his hand is away from the window.
Revealing mistake: When Indy's car is about to crash against Mac's, there's a wide angle where the car is seen totally black inside, revealing it's empty.
Continuity mistake: Just before Irina enters the warehouse with her men you can see she has no sunglasses on, as she took them off while speaking to Indy, but as she enters the warehouse we see her remove her sunglasses and hand them to one of her men.
Continuity mistake: The CCCP emblem on the back of Irina Spalko's uniform appears and disappears between shots, most notably when she is descending off the cliff and we get a clear view of her back-minus the emblem.
Continuity mistake: As Indy's truck is getting closer, the camera focuses on Spalko. Behind her are the duck with Oxley and Mac inside and a damaged truck. But in the next shot, when Oxley says "Henry Jones Jr.", the truck is gone. (01:16:17)
Continuity mistake: Before being knocked by Mac, a Russian soldier throws the Skull to Spalko, who has her rapier in her right hand. In the next shot, the rapier is in its sheath. (01:18:48)
Continuity mistake: When Spalko is trying to throw Indy's car off the cliff, the knocked-out soldier in the passenger seat of the amphibious car is suddenly conscious and sitting between shots. (01:22:30)
Continuity mistake: After Mac knocks the Russian soldier out of the car, the position of Oxley's head and arms changes in the next shot. (01:18:49)
Continuity mistake: When Indy and Dovchenko are shot out of Hangar 51, the expression on the Russian's face changes between shots. (00:17:00)
Continuity mistake: After Mac betrays Indy again, and Indy asks him if he's a triple agent, Dimitri suddenly appears next to Mac before Spalko enters the room.
Continuity mistake: While the soldiers are loading the coffin inside the jeep, you can see Spalko on the left side of the jeep, handing his rapier to Dovchenko. In the next shot, she is next to the back of the jeep with the rapier again and repeats the handing action. (00:11:39)
Continuity mistake: The Russian convoy consists of one staff car, two jeeps, and three trucks. When the convoy arrives at Hangar 51, a second staff car has appeared, from which Irina Spalko steps out. (00:02:44 - 00:05:50)
Continuity mistake: While the staff car and the black car are racing, there's a wide shot of the place, with only the road and some mountains in the background. In a few shots later, the Atomic Cafe has appeared from nowhere. (00:01:52 - 00:02:20)
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.