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)
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.
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.
Revealing mistake: When the baddies are chasing Mutt's bike, right after they exit the cafe they skid to the left. Skid marks from previous takes are visible on the road.
Revealing mistake: At the end Indy picks up a heavy slab and uses it to knock the carved faces off the wall. He hands it to his friend Mac, then takes it back off him - at this point it's clearly a very light prop.
Revealing mistake: When Indy and Mutt ride on the bike in the college gardens, there's a wide aerial angle that reveals that the distance to the red banner is very short, yet in every following angle they're repeating the same route over and over again (3 times at least) to make it look way longer and increase the suspense.
Revealing mistake: During the campus chase scene, Mutt skids his bike and turns left to drive under a stone arch. Multiple skid marks are seen on the pavement from previous 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.