Very, very rarely should you get an issue with the standard flow mesh, even the cut edges, causing bag punctures.
Even if so, a quick bit of bag tape over the affected area, which should be quite visible, will fix the problem and any air should evacuate on all but the smallest of parts
I'd check that the leak isn't, in fact, coming from the edge of the part. A pin-hole leak will show itself as a clearly defined trail of bubbles originating from a single point but, if the leak is coming from the perimeter of the part - from a pleated area in particular - the visible signs will be much more widespread over the part
Sometimes it's just a matter of getting your process nailed-down and it's not uncommon, if the bagging-up is a little untidy, for leaks to become apparent only when you start running the infusion.
Ensure that your pleats are well positioned and of sufficient size to allow plenty of slack in the bag and keep them to the minimum required to achieve this i.e. the more pleats you have then the more likely sources of a leak. I'd favour less big pleats over more small ones.
Also, when pulling down, take time to position the bag so that it takes a little strain away from the pleated areas. The pleats will have a tendency to pull inwards when vacuumed down, so back off the vacuum a little and position the bag so that the pleated tape sits as upright as possible when the bag is fully pulled down. This will only be possible if you've made the bag big enough to cover the mould properly without being over-stretched. Once you're happy with everything and it looks like you've achieved an airtight bag, go around the entire perimeter and give the tape a final press down to ensure that all is firmly stuck down, paying particular attention to your pleats as these will be the source of the majority of air ingress.
A little 'doughnut' of tape around the top of your silicon connectors can also help ensure that the bag doesn't wrinkle-up here when you insert the resin and vacuum lines, which can help minimise the risk of another potential leak source. Giving the perimeter of the mould a good clean-up with some of the mould cleaner or acetone can also be a good idea (only the space where you intend to apply your tape, though) as this will ensure that the tape sticks down well and isn't being slightly loosened by your release agent.
Stick with it as you can find yourself pulling your hair out in the early stages and it can take a bit of time to get yourself into a neat routine but you will get to a stage soon where mishaps become a rarity