Wouldn't trust my bike in a plastic bag TBH, especially with baggage handlers.
The only time I've flown with my bike I used an
ALAN bike box. I hired mine though from a local LBS.
Lots of people just use cardboard bike boxes and plenty bubble wrap though. You can pick one up from your LBS.
I am not being funny but having only flown once with your bike, you have not got a lot of experience to fall back on.
I have flown all over Europe with my bikes and find that baggage handlers take care of bikes. The only problems have had have been of my own making and always concern mudguards.
I carry mine bike in a home made bag, made from spinnaker cloth. It is very light and thin material, but strong. I turn the handlebar, take off pedals but put them back on the inside. I take off the derailleur and fasten it to the bike and just let it hang. I take the caps off the valves but don't let the tyres down unless I am told to. I have never had a tyre go POP or a bike damaged or lost.
I have a long, thin length of elastic chord that I wrap around the outside of the bike bag and it gives the handlers something to grab hold of.
When I reach my destination the bag folds up small and goes in a pannier. That way I am don't have all the hassle of dumping boxes or finding another box to come back. I simply ride to and from the airport.
I have read very few instances where bikes are damaged in transit. I cannot think of one where the bike was made unusable.
A lot of people on here uses C2C bags with tape and seem to manage OK.
Steve