In my experience, If they notice that it's a bike, or the size of a bike, or just the box of a frame, you will have to pay the extra for transporting bikes. At least in cheap flight airlines, that is how I fly. But, the cost is fair, about 40-50€, and they allow between 30-40Kg. What I do is include lot of clothes and stuff. Sometimes I pay just for the bike, not luggage, so I include clothes with the bike, so i save money and I can carry lot of weight. I always use a cardboard box of FRAME, for travelling with my bike. It's quite strong if you pack it properly. Only issue is the lack of wheels, but next time i will create a wooden deck with industrial wheels to attach to the cardboard box. No issues so far.
Happy Trails !!!