How can I convince my family to approve of marrying a girl who is originally Shiite but converted to Sunni, given that they know of her religious commitment, but are concerned that her family is still Shiite?

Among the characteristics of faith is helping a Muslim to remain steadfast in their religion and in obedience to Allah, for Allah says: (And cooperate in righteousness and piety).

One must be cautious that Satan might open the doors of temptation under the guise of helping a Muslim woman, as this could lead to falling into what is forbidden.

The family's refusal to marry her due to her family being Shiite might be justified, because in-law relationships are strong connections that require interaction and love, and this might be difficult with fanatics.

However, if your family is convinced by the girl, and her family agrees to the marriage despite the difference in madhhab, this indicates that they are not fanatics, which opens a door for dialogue and attracting them to the Sunnah, or at least establishing a respectful relationship.

This is not impossible, and one can seek the help of wise individuals to discuss the matter with the family, while clarifying that the girl is not to blame for her family being Shiite, and that Allah will help her and keep her steadfast if she is sincere in her transition to the madhhab of Ahl al-Sunnah.