Is the relationship between a young man and a young woman—which began with online acquaintance and developed into meetings and a marriage proposal with the mother's knowledge and intention of marriage—considered haram (forbidden)? What is the ruling on promising not to marry anyone else? What is the interpretation of the noble verse: "And do not promise them secretly" in this context? And what should she do regarding this relationship?

Such a relationship between a girl and a non-mahram man is forbidden in Islamic law and is not permissible, even if her mother knows about it or if there is a good intention. It must be severed, and one must repent to Allah. Furthermore, a girl should not promise a young man that she will be his wife to the exclusion of others. The verse mentioned in the question from Surat Al-Baqarah pertains to proposing to women during their 'iddah (waiting period). The best guidance for two people who love each other is marriage; if it does not materialize, then Allah's bounty is vast.