Kids typically come home, with a homework folder or school planner. In it, daily as a class, they are supposed to write in it, what is homework and what is coming up. Also, there will be papers for the parents as well, sent home in that folder.
Does she have this?
My daughter is 9 and my son is 5, they are both in school and I check their homework folder daily.
So I know what is coming up and what they need to do.
If your daughter is not coming home with a homework folder/homework nor is writing it down, then you need to also ask the Teacher, what is their routine, daily, per their assignments. And tell her what the problem is.
In 4th grade, it will be more responsibility. And the kids are expected to be more independent. Per their taking their homework home, writing down their assignments and bringing home what is needed.
**Adding This:
Your daughter's problem, seems to be that she does not know how to organize herself and seems to "forget" homework.
Organization skills, for some, are learned. And taught.
And, every classroom has a routine to it. If your daughter cannot follow direction or know the classroom routines by now, you need to speak to the teacher. And ask the teacher IF homework is given daily. Usually it is. So "no homework" is uncommon.
But if your daughter is "lying" about homework, that is another issue.
To me, her problem is organization. And consciously not keeping track of homework. Maybe just being "lazy."
She already gets A's and B's... so, it seems... she can do the work... if she is keeping up and studies. But so, perhaps a "Tutor" is not the answer. Because academically, she seems to get A's and B's. Meaning she can do the work. Academically. Her issue is: she is not organized, and "forgets" and seems lazy, to keep up to date. And does not study as she should.
Study skills... is her other issue. Thusly.