Check my gallery, fair number of BiF. Granted, these aren't DSLRs, but if you wanted that level you wouldn't be asking about a superzoom bridge camera. What does help is get the fastest memory card the camera will support, it helps with sustained bursts. BiF are easiest with "spray and pray": track the bird and push-n-hold the shutter button while following.
Not sure what you mean by "it can take only 300 pictures." I don't know of any modern digital camera that is limited like that; you're limited only by the size of the memory card. I routinely take 1200+ photos on a single card (JPEG, max quality, lowest compression).
If you mean batteries, then you might be getting close. I would say the weather greatly affects this. Warm weather I can get 900 photos or so on my Vivitar batteries (slightly more capacity than the stock battery), but when it's cold that can drop to 300 photos. My record low was barely 200 shots because of cold weather (about 45°F). I always carry at least one spare battery on all my cameras for this reason (right now I carry one on the camera and two spares).
By comparison, my GF's D700 can go alot more photos per charge, but her built-in battery is quite heavy and isn't using excessive energy running an EVF. But hers go alot faster when it's cold too. And her RX100 pocket cam is only good for about 200 shots on a good day before the battery dies (tiny battery and power-hungry electronics).