It is said that Fatima al-Zahra (A) and Imam 'Ali (A) got married in the second year after the migration to Medina, after the Battle of Badr (so approximately 624 AD)
However, biographers do not agree about when Fatima al-Zahra (A) was born. Sunni biographers often say she was born around 605 AD, whereas Shi'i biographers often say she was born around 614-615 AH. There are some other views as well.
So, she would have been around nine or nineteen years old at the time of her marriage.
It seems reasonable that she would have married young since men are eager to seek the daughters of important men for marriage, and it is reported that others had already asked for her hand. But God knows best.