My littlest baby is now 13 weeks and we have just been in the trenches with the newborn sized modern cloth nappies (MCN), prefolds and fitted nappies. Even as a seasoned cloth nappy user (this is baby #3) there were times when our nappies leaked with our newborn.
The reasons were often:
Clothes accidentally tucked into the nappy.
With us it was singlets that were often the culprit here. If any clothing touches the absorbent materiel in the nappy then wetness will wick up onto the clothing creating a wet patch.
The absorbent part of the nappy poking out.
If any of the cotton, bamboo, hemp is poking up out of the cover it will draw moisture from the nappy up onto the surrounding clothes and create a dreaded wet patch.
A penis pointing up instead of down
Sounds so simple but quite often this was the reason we had leaks (pointing finger accusingly at husband).
Nappy not on tight enough
Cloth nappies need to be fitted on nice and tight. The elastics should be on snug around the legs and waist. If you see gaps there may be leaking. Elastic marks on newborn skin is normal.
Babies legs are too skinny
Sometimes it is hard to get a good snug fit around baby's legs if they are particularly skinny or if your baby is brand new. If you put the nappy or cover on the tightest settings and there is a gap, then there isn't much you can do until your baby chubs up. I found that my nappies were still mostly functional, even when there were gaps with the nappy on the smallest setting. Flat nappies, prefolds and fitted nappies can work best in this case, as you can fold the nappies and then add a cover giving two levels of leak resistance.
An absorbency issue
Unless your baby is a heavy wetter, I would expect that you could change their nappy at every feed (unless poohed in earlier). If baby has a really big sleep or isn't changed regularly enough then the nappy may leak.
The way to check if absorbency is the issue is to feel the inserts or soakers when you take off a leaking nappy. If the entire absorbent part is sodden, then your baby has out wet the nappy. You could change more regularly or boost your nappies with extra absorbency.
Your baby is a heavy wetter
Newborn sized MCN's are tiny, and utterly adorable, but this means that they have a limited amount of absorbency. Some babies are super wee-ers and may need to be changed a little more frequently or nappies may need to be boosted. Often fitted nappies, prefolds and flats are better for our super wee-ers as they have more absorbency. Heavy wetting newborns are often fine with larger sized nappies or One Sized Fits Most nappies.
I find that using cloth nappies are trickiest to fit with newborns with their skinny stiff little legs. Remember, if you aren't having much luck... it does get easier!
There is plenty of help out there. Most nappy makers and sellers (including me!) are happy to help troubleshoot any problems.