Although House Lannister is the new royal house, following Cersei's ascension to the Iron Throne, the opening credits still have the stag sigil of House Baratheon over King's Landing.
When the Hound and Thoros are filling up the graves, you can clearly see the shovels are modern day shovels with the demarcation from metal spade to wooden handle.
Jaime says that the ironborn broke their promises and murdered their former friends (namely the Starks) as soon as it suited them. This is incorrect, since the ironborn have never promised anything and never formed any alliance with the Starks.
Jaime asks Euron "Weren't you the one who started that rebellion by sailing to Casterly Rock and burning the Lannister fleet?". According to the novels, and also according to Tyrion (Cripples, Bastards, and Broken Things (2011)), the destruction of the Lannister fleet took place in Lannisport, not in Casterly Rock; it was not Euron but his brother Victarion who led the raid (though the plan was Euron's).
When Sam is collecting dishes from the feeding of the sick men in The Citadel, the heavily scaled arm that appears, and the voice enquiring about Danaerys' arrival belongs to an unknown man. English subtitles however give away his identity as Jorah before this is later confirmed.
"Dyre Den" is misspelled as "Dire Den" on Cersei's floor map.
Building a "1,000 ship fleet" with medieval level technology would have taken many years, if not decades. More importantly, many of the processes, such as curing freshly-cut wood takes time that cannot be made to go faster regardless how much labor is applied to it.
Cersei tells Euron that he has broken promises to allies and murdered his brother. Euron has indeed killed his brother Balon, but there is no mentioning, neither in the books nor in the show, that he has ever betrayed any allies.