It is possible that this question is asking about several different times in the book because Karana does try to leave the island twice and succeeds in leaving on the third attempt.
The first time that Karana tries to leave the island is when the remaining villagers leave the island in hopes of making a better life for themselves on the mainland. She wants to leave because she wants to stay with the people she knows. Unfortunately, Karana's brother doesn't make it to the boat on time, so Karana jumps into the water in order to stay on the island with her brother. Her brother then dies shortly after, and she is left on the island alone.
It is unbearably lonely on the island and filled with all kinds of painful memories. Karana wants to leave, so she makes an attempt for the mainland in a leaky canoe. The attempt fails, and she is forced to return to the island. She remains on the island for years, but she will eventually leave once more at the end of the book, and readers are left to wonder whether or not her life will actually be better.
I thought of Rontu lying there beneath the stones of many colors, and of Won-a-nee, wherever she was, and the little red fox that would scratch in vain at my fence, and my canoe hidden in the cave, and of all the happy days.