Nice card game! It was fun to see what cards you get to play and also what kind of enemies show up.
Not sure if I'm simply not playing my cards right, or choosing the right upgrades but I did find myself stuck at stage 4 due to lack of damage. The enemy bird would simply keep dodging my attacks and healing back up to full after I've managed to bring it down to half health. I kept it up for about 7 minutes before I eventually gave up with the loop. Perhaps you could have the enemy AI choose less optimal cards the longer time passes?
Either way, it was well done and very fun to play. Excellent work! :D