Lady Lady!! (1988)
When Rin's mother dies, she makes the promise that some day she will become a Lady, and travels to England to meet the rest of the family. She is not very welcomed (specially at the start by her stepsister Sara), but after some time they are all a happy family. Rin (Lynn) is also very interested in horses and racing, and she wins a nice share of contests.
This series is divided into two parts, first with Rin being really a child, and the second with her being around 13.