David and the Phoenix by Edward Ormondroyd

With a tremendous surge of its wings, the Phoenix managed to seize a branch. David's legs slipped from the bird's back, and he dangled over the abyss.
Thus ends the near-disaster of their first flight together. But don't underestimate the Phoenix! Failure only makes David's new-found friend determinded to get into shape so that David's education for Life can proceed.
And get into shape the fabulous bird does just as they conclude their first successful (and very scary) adventure, the Scientist appears. Don't underestimate the Scientist, either. He is just as stubborn as the Phoenix, and is driven by one obsession: to become famous by capturing the mythic bird. And if his traps don't work, he is fully prepared to shoot the Phoenix...