This "sleep regression" that happens around 3 or 4 months old is not a "regression". The word regression suggests that if you do nothing, it will get better all by itself in time. This is not true.
At around 3-4 months old, as baby leaves the newborn fourth trimester phase, there is a permanent change in sleeping physiology and a permanent increase in calorific need.
During the first few months of a baby's life, being sleep is their natural state. It is passive, it doesn't require any thought or effort from the baby. If all is well (no hunger or pain), baby will be asleep as their natural state of being during those first few months - just as is the case in-utero. Waking happens only when a need is not met (food, warmth, pain, uncomfortable) and being awake happens for only short bursts. Therefore relatively speaking, getting baby to sleep is easiest in the newborn stage.
As baby leaves the newborn stage, so their sleep matures. Being asleep is no longer their passive, natural state of being - babies start spending more time awake. Now baby can be fed, warm, comfortable and not in pain - but stay awake. Sleep matures to have stages, the sleep cycles adults have, rather than a passive state.
This is the age, from 3 or 4 months, that for the first time a baby needs to learn how to 'get to sleep'. Getting to sleep (and staying asleep between sleep cycles) is now active and requires effort, rather than sleep being a passive thing that requires no effort.
Babies who have established mechanisms in place to get to sleep even while they were newborns will be in a better place to now be able to actively get to sleep. Otherwise, parents now need to introduce these mechanisms.
Babies can be actively encouraged to sleep by using
- movement (bouncing, rocking, walking, car, pushchair)
- sucking (dummy, breastfeeding, finger, sucking a toy or blanket)
- Tight, secure feeling (holding in arms, swaddling, snoozepod/sleepyhead type of mattress.
- full tummy also helps baby get to sleep and stay asleep.
I have always favoured dummy and bouncy chair for daytime sleep and dummy and swaddle (in a side-car cot) for night time sleep - with the long term view of independent sleeping that does not require my input for sleep.
As well as sleep, milk intake also significantly increases at around 3-4 months old. Again this is not a phase, it is a permanent change. My DD's milk intake went up by 25% at 12 weeks. That's no small amount. She had both more milk per feed and also added additional feeds by making her full feeds more frequent.
My DC4 changed to FF at 13 weeks old during this change, but I have BF through this change in my older children. It does require significant hard work and difficult days to increase supply. But as you will have been told from your HV, the answer when breastfeeding is simply to put baby to breast as much as possible to increase supply.
What can I do?
Accept that you will now need to work hard to get baby to go to sleep.
Good sleep promotes more sleep. So the more daytime sleep baby has, the less over-tired she will be and the easier to get to sleep at subsequent naps. An over-tired baby is hardest to get to sleep.
Watch awake time, not asleep time through the day. Short naps are usual at this age - 20-45 minutes. The next sleep development happens around 5-7 months old when baby learns to sleep through one sleep cycle into the next - until then daytime sleeps are likely to be one sleep cycle, so 20-45 minutes.
So while naps may be short, keep them frequent. I would expect about 60 minutes awake time between daytime naps at this age, 90 minutes from waking at most. If baby takes a long time to get to sleep, then the playtime between naps may get even shorter.
Making daytime feeds frequent also helps to 'calorie load' through the daytime, helping reduce the need for night time sleep. I would go for really frequent feeds, especially so if you are BF and trying to increase supply. Even if FF I would go for 2 hourly daytime feeds.
Then be consistent with your night time wake ups. Dummies are great if trying to resettle without a feed, if you are daytime calorie loading. I also cant recommend enough a 3-sided side-car cot so that you can cuddle right into the cot to sooth and settle baby without picking her up.
Try to avoid needing to pick baby up and rock, because this is one of those awful 'rods for your own back' you may hear about. Settling baby to sleep in the cot at night is best long-term.
But if you are breastfeeding then safe co-sleeping and free access to the breast may be your saving grace through the night.