My 12 week old sleeps in a mesh side bassinet in our room, he constantly is turning his body in his sleep and ends up almost horizontal with his head right smooshed up against the side of the bassinet. Is there a way to prevent this? And with it being mesh is it safe to assume he can still breath just fine? I wake up constantly to check he is breathing and am forever repositioning him.

Are you still wrapping/swaddling your son?
It sounds like he is attempting to roll, so the safest environment for him now would be to move from bassinet to a cot, & to cease swaddling.
This is not something that needs to be stopped, but it is important to keep making his sleep environment as safe as is possible.
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
Red Nose recommends to move baby to a cot & cease wrapping once baby is attempting to roll.

Give baby extra tummy time to play when awake and supervised as this helps baby to develop stronger neck and upper body muscles which in turn enables them to roll back over.
Make sure that baby is on a firm and well-fitting mattress that is flat (not tilted or elevated).
Make sure that baby’s face and head remains uncovered (do not use lambswool, doonas, pillows, cot bumpers or soft toys)

As babies grow and develop they become very active and learn to roll around the cot.
At this time, continue to put them on the back at the start of sleep time, but let them find their own position of comfort.
By this stage it is not necessary to wake during the night to turn baby over to the back position

Do not use any devices designed to keep baby in a particular sleep position.
https://rednose.org.au/article/what-do-we-do-now-that-a-baby-has-started-to-roll-over