Continuity mistake: During the mine cart chase, Indy hits one of the switches, putting it from the forward to the back position. When the guards in the mine cart get close, they shoot it, putting it from the front position to the back position again.
Continuity mistake: As the Thugee ceremony comes to an end, you see the stage hands exit and the spectators get up and start to leave. The next scene shows Short round looking over the edge and the stage hands are still on the stage, just starting to leave and some of the spectators are still in the process of standing up to leave.
Revealing mistake: After Indy escapes from the conveyor belt, he runs on some suspended planks while being shot at. Watch the planks: There's black masking tape where the explosives are (which disappear in later shots). Furthermore, right before he takes out his whip, he grabs a railing where the bullet has just hit, and you can see that the black tape is dangling loose after the spark.