‘And Abram said to Sarai, “Look, your maid is in your hand. Do with her whatever is right in your eyes.” And Sarai treated her harshly and she fled from her face.'
Abram passes his judgment. Sarai is given authority to act as she sees fit. The woman is still her maid (it may be that this is an intentional downgrading of Hagar who had become more than just a maid). Whatever she does will be seen as having his sanction. He accepts her, in accordance with custom, as still the principal wife. Hagar possibly did not understand that Sarah was unique as a child of Terah, thus being of the tribal aristocracy.
Sarai then makes clear her position to the tribe, who will have been watching the power struggle and waiting to see what Abram would do, by her harsh treatment of the slave who has tried to rise above her station and who has responded badly to her mistress's kindness. This also was in accordance with custom. In the code of Hammurabi the punishment for a servant girl who bears a child by her master and seeks to take advantage of the situation is that she be reduced again to the status of a slave.
The harsh treatment does not necessarily involve unfair treatment, it lay in the downgrading that necessarily followed with all that that involved. But Sarai was human and felt she had right on her side, thus it is probable that Hagar had a very hard time.
Hagar cannot accept her new lack of status or her treatment and flees in the direction of Egypt, her homeland. In many ways she had given Sarai little choice. (One of the things that is said to cause the earth to tremble is ‘a handmaid who is heir to her mistress' (Proverbs 30:23)). Her attempt to supplant her had had to be treated harshly in order to re-establish Sarai's overt authority.
Of course her flight exacerbates her wrongdoing. She has no right to leave the tribe and she has not been turned out. Had she stopped to consider earlier none of this would have happened. She must have known the customs, even though as an Egyptian she was unwilling to subscribe to them. But she had made a bid to rise above her station and the consequence of failure was inevitable.
Yet the narrative is very sympathetic to Hagar, even though according to every custom she was in the wrong. In the light of the fact that the covenant it witnesses to, and establishes, is with her and her seed, it is clear that it was written by a sympathiser in the tribe who records it for her at Abram's request (the whole narrative reveals what a strong minded woman she is).