The special effects crew failed to rotoscope King Ghidorah's rays when the huge bridge is destroyed. As the shot now plays, the bridge is destroyed by a series of explosions of unknown origin when King Ghidorah simply flies over it.
On the US Naval ship in 1944, the Captain refers twice to another officer as, "Major (Spielberg)". The rank of Major doesn't exist in the US Navy, and the collar device Spielberg wore was that of Lt. Commander.
The time-travel mechanics of this movie are infamously confusing. The plot involves the characters traveling back in time to stop Godzilla from coming into being. Yet when they come back to the present, everyone still remembers Godzilla, even though he's been presumably erased from history.
In 1944, the downed Godzillasaurus is teleported to the bottom of the sea to keep it from turning into Godzilla. It then does become Godzilla when it absorbs the radiation of a nuclear submarine in the 70s, but it's not explained how it remained alive for thirty years under the water.
When Ghidorah is jumping on a downed Godzilla during the fight scene, you can clearly see that Ghidorah's foot bends as he lands on Godzilla. This shows that he was made of some sort of rubber (which also explains the lack of leg movement as he jumps.)
The goof item below may give away important plot points.
Shindo cannot join the company of the time travelers because the same person cannot exist twice at one time. This "rule" however is later broken when King Ghidorah is killed and his reanimated future-self is brought back to the present, thereby causing two Ghidorahs to exist at the same time.