In the book and show, Jaime trains up his left hand for sword fighting, but in the show in season four he trains with Bronn at King's Landing. In the book "A Feast for Crows," he trains with Ser Ilyn Payne, the king's justice (the guy who lopped off Ned Stark's head) while on his way to break the siege at Riverrun (which is broken in season six in the show, much later than in the book). Jaime picks Ilyn Payne because the man can't talk (he had his tongue cut out). Jaime's biggest fear is that his training partner will tell others that he's a bad fighter now, but he doesn't need to worry about that with Payne. It's a smart tactic.