What is the Sharʿī (Islamic legal) opinion regarding the case of a daughter who entered the bathroom, got scared in it, and consequently started acting strangely and became afraid to enter alone, and what should be done for her to return to her normal state?

My advice is to fortify your daughter with remembrances (dhikr) and authentic supplications (du'a), and by reading the Qur'an. Strengthen her conviction in Allah, that none but Him can benefit or harm. Occupy her free time with beneficial readings or permissible hobbies. There is no harm in seeking a righteous ruqyah practitioner (raqī shar'ī) of sound belief, or in presenting her to a psychiatrist.