My daughter is a week away from being 5 months old & is still not sleeping through the night.
We have a routine, bath, bottle, bed around 7pm. She usually falls asleep fine until around 10pm where she will start to fuss and cry, I've started to let her self soothe as I know she isn't hungry. She will fall back asleep until around 12:30 and from then she fusses every hour or two. I give her a bottle somewhere between 12 and 3pm and then she is fully awake at 5:30/6ish.
When she wakes up fussing she is still in a sleepy state with her eyes shut so I'm not sure why she's wakening up🤔