Deliberate mistake: After Ramsey shows Sansa her flayed northern friend, when it cuts to Davos riding the horse through the encampment one of the shots is flipped. Note his sword and his cloak's crossed straps have switched to opposite sides.
Unbowed, Unbent, Unbroken - S5-E6
Continuity mistake: When the Sand Snakes attempt to kidnap Myrcella, as Nymeria Sand runs up to Myrcella to seize her, she is holding her large whip in her right hand. However, in the very next shot, the whip has disappeared, and Nymeria is now holding a dagger in her hand instead.
Continuity mistake: When the Faith Militant descend on Ser Loras Tyrell's training session there is a shot after the Militants have come down the stair of Loras fighting from behind a man's shoulder, in this shot you can see they are fighting on grass surrounded by walls with sheer drops with no entry point other than the stairs the militants walk down. In the next shot you can see the 9 militants that have come down the stairs approach Loras from one side, but suddenly 6 militants join him from the opposite side, where they definitely were not before. In the zoomed out shot you can see there is nowhere for these men to have come from. (00:11:15 - 00:11:50)
Continuity mistake: In the scene in the amphitheatre, while seated in the royal box, Daenerys is wearing a full length white gown with flat pumps and an ornate silver necklace with a large dragon motif on the left side. During the revolt when she is seen running across the fighting pit the dress is shorter and she seems to be wearing some sort of leggings underneath it. Daenerys escapes on the back of her dragon and when we next see her in episode 10 she is wearing a completely different knee length pale blue gown. The necklace also seems different with her hair spread out on her left shoulder to disguise the lack of the dragon motif. She is also wearing grey boots with a fairly high heel. Obviously dragons come with a secret baggage compartment.
Continuity mistake: Towards the end there is a body of a dead Son of the Harpy lying between the fallen Unsullied and the side corridor visible in the wall, located very close to either. In the next show the body is gone. Also, as the camera angle changes, the blood stains on the walls look differently. (00:48:50)
Answer: This is explained more in the book. Khal Drogo and his bloodriders do intend to help Viserys take the throne, however they do things in their own time. This is what causes Viserys's frustration. In the book, Illyrio Mopatis tells Viserys that he has already waited his entire life to claim The Iron Throne and that waiting a few more years isn't that big of a deal. The Dothraki are a nomadic tribe, which is largely why they have no interest in invading and sacking King's Landing. However, since Khal Drogo eventually grows to love Dany, her assassination attempt by the Westerosi compels him to seek revenge.