I put my 5 week old baby to sleep on his back, towards the bottom of the bassinet, in a safe sleeping bag, arms in, pulled down tightly on the shoulders and it appears well fitted when he goes to sleep. However, a few times i have woken to find the sleeping bag bunched up over his mouth, nose and ears, almost to his eyes. Is this considered ‘safe’? Do you have any suggestions? He is of an average weight, wearing the smallest bag, which is his size. The bag is a well known popular ‘safe’ brand. He has done this with bags of different tog ratings. He does tend to move around (moving arms and legs, moving whole body side to side) quite a bit even in his sleep, I am not sure if this is normal, I have been told he is quite strong. Thank you.

Red Nose recommends that baby’s head & face remain uncovered during sleep.
It is particularly important that nose is not covered, as babies can only breathe through their nose (obligate nose breathers)

It is not unusual for some babies to wriggle & move around their sleep environment before they are rolling.
Some babies do not respond well to wrapping & may wriggle to attempt to get out of the wrap/swaddle.
Many babies like to touch their face or suck fingers when sleeping.
Red Nose recommends using a muslin or light cotton for a wrap.

He may be wriggling/moving a lot because he is resisting the wrapping.
You could try wrapping him with his arms out - as with a baby around 3 months of age. (or use a sleep bag with arms out)
It is important that baby’s head & face remain uncovered & as you are doing, always place baby on the back for sleep, with feet at the foot of the bassinet.

If you wrap your baby, consider baby’s stage of development. Leave arms free once the startle reflex disappears around 3 months. Most babies eventually resist being wrapped. Wrapping style should be appropriate for the baby’s developmental stage.

It is essential to discontinue wrapping as soon as baby starts showing signs that they can begin to roll, usually between 4-6 months of age but sometimes younger. Babies must not be wrapped if sharing a sleep surface with another person. Baby should not be wrapped while sleeping in a baby sleeping bag.
https://rednose.org.au/article/is-it-safe-to-wrap-swaddle-my-baby
https://rednose.org.au/article/wrapping-babies

Wrapping & swaddle products
............. it is extremely important to ensure that the product fits the baby and is appropriate for their developmental stage. For example:

The material of the wrap or swaddle should not cover the face or head, particular if baby sleeps with arms in different positions. If the item is too big for the baby, some zipped swaddle suits that enclose baby’s hands, have been shown to allow material to cover baby’s face and nose when baby raises their hands above their head during sleep. All sleeping attire designed to cover the baby’s shoulders should have separate neck and arm holes or should ensure that they do not allow the face covering if the baby was to move their arms in different positions.
https://rednose.org.au/article/wrapping-babies

Depending on the size of the bassinet, he may be more settled in a cot.

Red Nose does not recommend the use of sleep positioners or anything that restrains a baby in a specific position.
https://rednose.com.au/article/is-it-safe-to-use-products-designed-to-keep-baby-in-a-particular-position

I used to tuck the bottom of my sons sleeping bag underneath the matress of the bassinet and his weight would keep it from untucking. Maybe see if this helps?