Question

why the gravity of earth do not works on on a small flying bird

Answer 1

So, the bird is "small enough" to overcome gravity. They have hollow bones and feathers that make them light, and when they flap their wings, they make enough force to surpass gravity, and that's why they can fly

Answer 2

Flight in birds and planes works in a similar way; both have to generate lift to overcome gravity and thrust to move forward in the air – but birds don't have engines to provide thrust! Instead birds flap their wings, which pushes a bubble of air (called a vortex) behind them and downwards.