Is a five-year relationship permissible between a man and a woman, intended for marriage, with the knowledge of the girl's family, and involving conversations that are not devoid of romantic talk, yet free from indecency?

Such communication and exchange of conversations between a man and a non-mahram woman are not permissible. An emotional relationship between them is religiously forbidden, even if it is devoid of passionate love. It falls under the category of "khidan" (secret lover/friendship) which is prohibited in Islam, as it involves following the footsteps of Satan and falling into forbidden acts. Conversation between them is only permissible out of necessity and within the limits of Islamic regulations. Repentance from such a relationship is obligatory. The desire for marriage or the families knowing about it does not make it permissible. If you desire the safety of your religion, then cut off this relationship. There is no harm in striving for marriage; if it becomes easy, then praise be to Allah, otherwise, leave her and look for someone else. The cure for passionate love is possible.