Continuity mistake: When the thug jumps on Indy's mine cart, Indy's hat gets loose, but in the following shot it's perfectly tucked in.
Continuity mistake: After Shorty is saved from the thugs' mine cart, a baddie jumps from an elevated cart situated on the right side, but when the angle changes he is coming from the left side. Also, Indy is looking sideways instead of upwards, as if the bad guy was parallel to him.
Audio problem: In the mine chase scene, when both carts are parallel to each other, Indy punches a thug. A frame later, from a different POV, two quick punches are heard, but no one is punching anybody.
Continuity mistake: When the assassin's head is about to hit the fan, in the first shot of him going up, his head is very close to it, but in the next shot, it's further away.
Continuity mistake: When the Indian guest is given a beetle to eat during the dinner sequence, he pulls off the back of the insect. When he notices that Willie is not eating, you can see that he is only holding the beetle, but when the camera angle changes in the next shot, the guest is holding the insect with the part he pulled off next to it.
Continuity mistake: When Indy throws a flaming skewer at the Chinese guy, the right side of his scarf runs down to his waist, yet in the close-up it's shorter and stops by the ribs, check out the black lettering embroidered on it.
Revealing mistake: When the 3 heroes exit the mine cart and escape from the water flood in the mine, Harrison Ford is replaced by an obvious stuntman.
Revealing mistake: When the water tank falls over, the vapour fumes are moving in slo-mo, despite the whole scene moving at a normal speed.
Continuity mistake: When Mola Ram tells the Thugees to get to the huge cauldron, there's big clouds of smoke all around it, which disappear a frame later from a different angle.
Continuity mistake: When Indy throws a block of wood on the rails, it lays parallel, yet a frame later, from a different angle, it lays sideways.
Continuity mistake: When the mine cart is about to stop, Indy's left hand is alternately covered by Willie and Shorty's or not, depending on the angle.
Continuity mistake: When Indie and Willie are about to jump out of the bar's window, there are some cables along the window which disappear when they crash through it.
Continuity mistake: When the three heroes arrive at the village there's not a single bit of grass around. When they leave the place the morning after there's vegetation everywhere.
Continuity mistake: When the heroes leave the village, the shrine where the Indian is standing swaps from having a blue sky to a dark grey one, and smoke from a campfire all over.
Audio problem: While Indy and Short are trapped in the room with the spiked ceiling, Indy takes his hand out through a hole and grabs Willie's hand by surprise. A loud cry is heard, but she doesn't mouth it nor reacts with surprise.
Continuity mistake: During the hanging bridge scene, the brown mountain has some white rocks on the right side which sometimes disappear, or even appear on the opposite side as when Mula Ram throws the Thuggee down.
Continuity mistake: A man takes the wooden blocks off of Laoche's plane wheels but disappears in the next angle.
Continuity mistake: When the Maharajah enters the palace for the first time he is escorted by two guards. A frame later the guards are gone from behind and are already formed along the aisle.
Continuity mistake: The movie starts with a guy playing a gong, holding its hammer from the lower part. A frame later his upper hand is holding the hammer from a much higher position.
Answer: This is conjecture, but it seems the general function of setting Temple of Doom before Raiders of the Lost Ark is that it helps set audience expectations that the two movies are self-contained episodes. For instance, Karen Allen has said she wasn't disappointed about not being asked to return because she'd already been told that the next installment was being set in the past before her character is reunited with Indy. Conversely, since we're already aware Raiders makes no mention of the events of Temple of Doom, we know we shouldn't necessarily expect any further installments to continue directly from prior movies' storylines regardless if they are set forward in time.
TonyPH