At that early age, they don’t settle on their own. I found that cuddling and wrapping them snuggly in a swaddle helps them settle and sleep for longer.

If bottle-fed, just like my eldest child, try using air-vented baby bottles as they help reduce gas which is a common culprit for their fussiness. I used one from Pigeon https://www.pigeonbaby.com.au/collections/anti-colic-bottles-and-teats

None of mine really self settled at that age. Generally I fed them to sleep or they fell asleep in the pram if we went for a walk. If all else failed I’d put them in the car and take them for a drive.

play white noise in minimal sound. they will settle.

Just one hand on there belly and shhhhh . Start putting them back down awake but sleepy

Some babies just cannot be settled until they are older. It is not your fault. Tresillian have some great tips if you call, visit, or look at their website.

I was always taught that bub should be put back into their cot awake after a feed and you settle them there. I found this worked best for me and sets them up for a good routine in the future too. I never had any issues with getting my kids to sleep through, etc.
You simply put them in the cot, pat their back, hum, etc. If they are not settled after a few minutes, leave the room and allow them to fuss for a bit, then come back and repeat. It is important to let your baby learn to self-soothe.
Also, if you have done absolutely everything and they still won’t settle, there is nothing wrong with putting them in their cot and walking away. We had to do that with my son or never sleep again. He survived and he would always go to sleep on his own eventually. He also turned into the best little sleeper.

I’ve found that stroking their forehead very lightly and their head helps calm and soothe them plus having soothing music helps too. There’s music that guarantees your baby will fall asleep in 10mins on YouTube. I have never had to use it but maybe it’s worth a try. My daughter had special meditation music for my grandson and it worked a treat every time. Even as he got older it still worked.

Not all babies that young can resettle themselves yet. Our little guy loves his sleep sack (he’s just started rolling so learning how to have his arms out is the new challenge). I use a Sleepmac machine for light and sound to have some white noise running to block out other noises in the house. We have a serious contact napper and just have to roll with it. He’s only just started to nap independently at 3.5 months. Sleeps wonderfully through the night, but so clingy during the day. Game changer for me was a good nursing chair to rock him to sleep, then transfer to his cot.

In my opinion, dont overthink it or stress out. JUst cuddle your baby. I went through sleep training with my 1st after feeling pressure for others. it didnt work. WIth my second i just pretty much cuddled/held him whenever he was unsettled. so much easier. And its nice to remember those times and the bond you shared.

When my kids where younger I’d trade them on walks, I’d also like to put there beds out where I was but out of site of me as well so if they were to fuss I could here them and made sure the room wasn’t to light either

We liked to go for a walk in the pram.
At times a bath or massage woild work too.
And we cuddled our baby in our arms until they were tired or drowsy, placing them in their cot whilst they are still awake.

At that age, TBH, we mostly cuddled ours.

Wrapping them tightly in a swaddle.

A secure sleeping bag. One of my twins liked to hold my hand.

White noise in my daughters rooms worked as well as a black out blind, i think held my hand on her until she was a sleep in her bassinet and i sometimes rocked the bassinet slightly as they like that movement.

Creating a soothing bedtime routine and gently placing a comforting hand on your 5-week-old while they settle helps foster a sense of security and encourages self-soothing. Good luck!