Warfare isn't won by battle, but by logistics. First, you have the common goal of four mamailian (assuming the Merfolk are standard Little Mermaid fish flippers for feet type and avians are angelic). If we were to assume each faction can field equal armies, you are dealing with a four on one war. The current tactical might of the United States, the best military in history, is such that the United States can fight a war on two fronts simultaneously. Dragons may be comparable... but that still leaves them very vulnerable to hit and run tactics that spread the Dragon to thin and leave them vulnerable to offensive actions at every front. Even if the dragons choose to attack one race first, the other three are now free to enter the unguarded dragon territory and take all the spoils for themselves. This will either ruin prime hunting grounds and other resources OR force the dragons out of their offensive (The battle of Gettysburg, for example was the end of a campaign for Lee to distract the Union by invading their territory.)
Of course, assuming that these dragons are intelligent, vicious, large, and long lived with tough armored scales and fire or other breath weapons. A flight of dragons over the battlefield is scary, but if you are doing hit and run and you will likely not encounter many as they are spread thin. But then we also have look at a manpower issue (pardon the word).
Dragons traditionally are old and and mature dragons don't reach their status for some time. Meanwhile, humanoid races are quite good at breeding fast and often and reach maturity much more slowly. Sure, a single Dragon might require fifty allied soldiers to kill, but it's going take a lot more time to replace that dragon that the losses incurred by a 50 man strong unit. Even if the unit and the dragon are both wiped out in combat, the net loss will be on the dragons side because the unit can be more quickly replaced with able bodied troops (In WWII, the United States lost tanks at a rate of 2:1 in combat against the German Panzer. But the United States was able to manufacture Tanks of a rate of 5:1. Unless a Panzer killed six tanks each, any action was a net loss. In addition, the United States used the tactics of "Hit them where they ain't" and fielded tanks in combat zones where Panzers were not).
In addition, Dragons are notoriously lazy, prideful, and stubborn, even to their own. A dragon army will have constant problems. "The Dragon King is clearly sending me to do this because he wants my treasure. Well, it's not my war... heck... if I can have the Dragon King's treasure, I'll help the food kill him. #NotMyDragonKing". Dragon land politics is probably filled with all sorts of back stabbing and bickering and infighting that would make Congress look like an institution of agreeable people.
In addition, all four races are significantly smaller and more mobile than a Dragon. It's easier to hide a group of Wood-Elves in the forest where they would be somewhat sheltered from a Dragon. Or in Caves. Having Merfolk could also open up avenues in that they would have advantages to dragons underwater (provided dragons were efective in such an environment at all) and could even find ways in a fantasy setting of setting up shelters and refuges for war wounded could return to the battle sooner. The avians would oviously be much more manuverable in the skies and could possible do scouting, while the desert elves... um... help me here, are they elves that live in the desert instead of the forest or... hmm... perhaps they are engineers or good at making really sturdy shelters or tunneling... yeah, develop those... Dragons may live in caves, but a fight in those limits their mobility, especially in fight or flight situations (A good success of the Vietcong against the United States was avoiding the bigger ticket problems (the helicopters and the air planes and the tanks) an hitting the troops when they had them at a disadvantage.... What? Not all my examples are U.S. centric... okay, fine... The Continental Army beat the British Army at the height of the Sun Never Setting by having better lay of the land, better command of available resources, and a more readily resupply of manpower. If you've ever looked at how Americans fight, they don't like to be fair. Also, this is why a lot of Pro-Second Amendment advocates do see an armed uprising against the best military in the world as perfectly feasible... there wouldn't have been a United States if it wasn't possible. waves my little American flag and plays the Star Spangle Banner Oh, incidentally, the national anthem is about the second time they did it.).
Anywho, its entirely possible to war with dragons to the extinction of them. It will probably be costly and hard fought, but most wars are when your aim isn't to fight fair but fight smart and above all, fight dirty.