In latitudes that have four seasons, springtime means the rebirth of nature. Flowers bloom and trees grow new leaves. So it is easy to imagine this idiom, which indicates renewal, as having a springtime association. After all, isn't "trying to improve one's behavior or attitude" (which is the meaning of this idiom) similar to a tree growing new leaves?
It turns out that is only part of the story. The reference to "turning over a new leaf" actually refers to a blank page in a book, such as a diary in which you write the story of your life. Long ago, book pages were called "leaves" because pages are flat and broad like the leaves of trees. A "new leaf" would be an empty page, not written on yet. "Turning it over" means that a fresh start is possible. An empty page indicates unlimited possibilities.
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