Does the Queen follow through on her promise to Hamlet? At what points does she seem to be still on Claudius's side?

By the time we reach act 3, scene 4 of Hamlet, our hero's behavior has been getting increasingly strange. Polonius, for one, thinks that he is mad from a growing mental disorder caused by his thwarted love for Ophelia. He convinces Gertrude to give her wayward son a good talking to. In the meantime, Polonius will hide and eavesdrop on their conversation.
Actually, it is not so much a conversation as a full-blown argument. All of Hamlet's pent-up resentment toward his mother for marrying his father's killer comes spewing out in a torrent of bile. "You've insulted your father," says Gertrude, meaning Claudius, not Hamlet's biological father. "Well, you've insulted my father," fires back Hamlet. He, of course, is referring to his late, murdered father.
And so it goes until Hamlet begins to accost his mother so violently that she starts screaming for help. From behind the curtain, Polonius also calls for help. This alerts Hamlet to Polonius' presence, only he does not realize it is Polonius until he has brutally run him through. He thought it was Claudius.
Gertrude is shocked, but Hamlet is unrepentant. After letting fly with another choice string of increasingly vituperative insults, the ghost of Hamlet's father appears once more. Gertrude cannot see it, so when Hamlet starts babbling away she concludes that he has completely taken leave of his senses.
But Hamlet tries desperately to convince Gertrude that his so-called madness is all just an act. He makes her promise not to let on to Claudius. She agrees. When she next meets Claudius, she does indeed keep her promise not to let her husband know that Hamlet's madness is feigned. However, the urgent and distressed manner of her speaking makes it clear that she is now firmly on the side of Claudius.
In fairness to Gertrude, Hamlet has really left her no choice. If he is genuinely mad then it makes sense to ally herself with Claudius. On the other hand, if he really is just pretending to be mad, then his behavior is pretty bad as it is, as illustrated by his unhinged ranting at his mother and his murder of Polonius. Either way, Gertrude not unreasonably seeks the safety and stability of an alliance with Claudius instead of hitching her wagon to Hamlet, who, at the very least, is a highly unstable personality.