looloo711, sorry to hear you're struggling. It is very hard isn't it? My DH said today why didn't anyone tell us? I said we wouldn't have believed them until we went through it.
Anyway, my situation evolved and deteriorated very badly over a few weeks. She has silent reflux (oh what an oxymoron that is its anything but silent omfg!!) and it wasn't until we managed to get the medication into her that things improved. She got to the point where she wasn't sleeping almost at all and screaming hysterically for hours inconsolably. After about 5-6 days on losec it started to turn around. She started smiling again. The poor little muffin was so miserable she had stopped smiling, gosh we had a dark couple of weeks I really couldn't cope. I can't describe how bad it was. Anyway she's been having losec now for 2.5 weeks and I have a different baby. Totally different. I can put her on her playmat or in her bouncer for 15 minutes. Wow. She smiles all day and wants to play. She doesn't scream hysterically whenever you put her down. This was my issue. Not the fact she wasn't sleeping because if she was awake and happy I'd be ok but she was awake and absolutely miserable and distressed. So I believe her reflux pain is under control and that's why she's generally much happier in herself. It's so beautiful to see as she was a very unhappy little baby.
Sleep-wise it's a bit better but not perfect. I've been intensively focusing on monitoring awake times and settling for naps. I can now get her to sleep 1-2 hours in the morning, try for 2 hours midday and 1-2 x 30-60 minute naps between then and bedtime which is 6:30-8pm depending on feed times (breastfed on demand). She still wakes up after each and every sleep cycle during the day which is frustrating but 1/2 the time I can resettle her. And at least she's getting some sleep. I am using white noise (rainfall) for all day naps, dark room, tightly wrapped in a Velcro swaddle and rocking the bassinet until she closes her eyes then leaving the room. Will be putting the cot on wheels for when I transition to the cot in about a month. If you wheel the cot/pram back and forth over a small bump (eg edge of a rug) creates similar movement to rocking. Maybe this is a bad habit but works for us and was recommended by child health nurse.
We did get a sleep consultant out and they recommend - strict routine and shush-patting but as much as I tried this it didn't put her to sleep. All babies are different. Mine did not like it.
One thing I've noticed is when she wakes in the morning for the day (between 6:30-8) she can only tolerate 60-75 minutes awake time then needs to go down for her nap. Miss this and likely to throw the rest of the day out as she starts off overtired. An overtired baby is MUCH harder to settle is the biggest thing I've learnt. For other awake periods she does 75-90 minutes maximum. I can pick her tired signs better - starts to grizzle and does a few yawns. I put her straight to bed. It's getting better.
For your LO - 4 weeks is still so early but yes it is hard. Use a wrap carrier if that helps - I swear by mine. And really monitor those awake times - an hour is probably enough. Dark room, swaddle, white noise and find a settling technique that works for your baby. That's all I can say. And hope that it gets better. My LO struggles with sleep that's obvious to me. I don't expect it to change but will work with her to help her sleep as much as I can. I wish she was one of those babies to just fall asleep wherever whenever (oh please like the pram eventually!) but no I haven't been blessed with one of those babies!! Take care hang in there and get help if it gets worse over the next month-two.