LOL, Charizard is Clearly Fire/Flying.
People who play the video game know this, but I doubt some random person who works in advertising for the trading card game (that doesn't even have a Dragon type) would know this.
Anyway, let's get technical, because I like arguing.
1. Charizard is in the Egg group Dragon.
2. In episode 105, Ash's Pokedex reads Charizard as the Dragon Rage Pokemon.
3. Charizard can learn the moves Dragon Claw and Dragon Rage without breeding. Through breeding it can learn Dragon Dance and Dragon Rush.
4. Lance, the trainer of Dragon type pokemon, owns a Charizard.
5. In Pokemon Ranger, Charizard appears as one of four prominently featured patron elemental dragon Pokémon.
6. Charizard's name in German: Glurak - Comes from Glut (ember, also the German translation for the move of the same name) and Drache (dragon).
7. Charizard's name in French: French: Dracaufeu - Effectively translates as Fire Dragon.
8. Charizard's name in Chinese: Chinese (Taiwan and Hong Kong): 噴火龍 Pēn Hǔo Lóng - "Fire spurting dragon".
So, Charizard IS a Dragon. He just isn't Dragon type in the games. He probably isn't because that would be unfair to the other Starter Pokemon that one gets a Special type. He'd also have some great weaknesses and resistances if he was Fire/Dragon!