There are two verbs here used to indicate Bathsheba’s behaviour as she came into the presence of King David. The ESV translates them as bowed and paid homage.
We do not need to see them as separate actions. The second verb is an explanation of the first, and it suggests that Bathsheba prostrated herself (laid face down) before the king. The Hebrew verb is also found in the second commandment that forbids the worship of idols, “you shall not bow down yourselves to them…”
Yet, in doing so, Bathsheba is not paying religious worship to the king, but she is showing the respect that subjects of a monarch gave at that time in the Near East. She gave David no less respect than he deserved, but neither did she give him more than he deserved, even though she was one of his wives. From a worldly perspective today, such requirements seem excessive and demeaning, but it was the norm in that cultural setting. It must be emphasized, nonetheless, that she paid homage, not because she was a woman, but because she was a subject.
David opened the conversation by asking what she wanted. The Hebrew phrase literally says, what to you.
We might see it as the same thing as saying, what is your concern,
or even what is your reason for coming to me?
The king’s question indicated his willingness, despite his physical incapacity, to hear what Bathsheba desired to tell him. It opened the way for her to speak.
16 Bathsheba bowed and paid homage to the king, and the king said, “What do you desire?”