Hi there, We are currently using a Moses basket bassinet for our 3 week old baby and have noticed that during the night she often moves and ends up resting her head on the side of the bassinet. Are we better off buying a bassinet with mesh sides so she is still able To breathe If her faces happens to become squashed against the side?

Maybe the bassinet is too small for her.
Are the sides soft/padded?
Some babies do wriggle/move at a very young age, & some may benefit by being transferred to a cot.
Are you wrapping her?

There are no mandatory standards in Australia for bassinets
Product Safety Australia & Ned Nose have the following information:
As there is no Australian Standard for bassinets, we are careful to keep up to date with reports of accidents associated with their use.
To date, the most frequent accidents associated with bassinet use are falls and suffocation hazards.
https://rednose.org.au/article/bassinettes

Ensure the sides of the bassinette are at least 300 mm higher than the top of the mattress base.
Ensure it has a wide stable base and a sturdy bottom so that it won’t tip over.
Ensure the mattress fits well and is firm, smooth and no more than 75 mm thick.
If the product has folding legs, make sure it cannot accidentally fold during use.
https://www.productsafety.gov.au/products/babies-kids/kids-furniture/bassinettes

We do wrap her but she wriggles all the time. The bassinet has a cover but no padded sides. We don’t currently have the space for a cot in our bedroom
So we will need the bassinet for a little longer. Would a bassinet with mesh sides be any better for her?

Are the sides of the bassinet higher than 300mm?
Are the bassinets with mesh sides any larger than the Moses basket?
Whether a bassinet with mesh sides makes any difference is debatable.

You could try wrapping your daughter with her arms out or cease wrapping altogether. Some babies wriggle a lot, as they are trying to get out of their wrap.
Have a look at the graphics on the information to wrapping baby, under principles of wrapping.
You could consider wrapping her as seen in the example of a 3-6 month old baby…...some babies do not respond well to wrapping from a very young age.
Examples of techniques that can be used to wrap a baby based on their developmental age. Reduce the effects of the Moro or startle reflex for a younger baby by including arms in wrap. Help an older baby stay on their back by wrapping their lower body, but leaving their hands and arms free to self soothe. Most babies eventually resist being wrapped.
https://rednose.org.au/article/wrapping-babies