Our 2.5 week old is in a bassinet at the moment (the sides are not mesh). We always put him down at the bottom centre of the bassinet, however when we get him up for a feed, we will find him close to the sides of the bassinet. He keeps kicking his legs and lifting one of his shoulders up to try and turn over, and his face seems to always end up about an inch away from the bassinet. Is it still safe to use the bassinet or should we transition to a cot as he is now finding his way to the sides of the bassinet that isn't mesh.

Many babies do wriggle & squirm & can be quite active & will move or roll around the bassinet.
Some babies do not respond well to swaddling/wrapping.
Sometimes it is because they like hands free to touch their face or suck on fingers.

Still always place him on his back for sleep.
Are you swaddling/wrapping your baby?
Are you are using a swaddle suit for a swaddle? ie arms are free or not?
Red Nose recommends using a light cotton or muslin wrap.
https://rednose.org.au/article/safe-wrapping-or-swaddling-when-to-stop-and-how-to-select-a-safe-sleep-bag
Some babies like arms free & will wriggle & attempt to roll especially to side to get arms free.
You could try sleeping him in a sleeping bag (arms free) or you could use a muslin type wrap with arms for 3-6mth old baby.

You could also move him to his cot. Babies can be placed for sleep in a cot from birth.
Red Nose does not recommend the use of any product that “restrains” a baby in a specific position.

Have you started tummy time when awake yet? How does he go?