To expand on the "breathing fire is inherently magic" point I'm going to throw out some "conservation of energy" arguments.
200 gallons of fuel is enough to power a "flame tank"'s weapon for 32 seconds. These things are designed to destroy other vehicles, not entire fortresses/cities such as Harrenhal, so we can pretty safely assume an adult dragon can throw out more fire than this.
Regardless, the young Dragons seen in the TV show demonstrate capabilities roughly equal to this.
200 gallons of gasoline contains 116,090 BTU of energy (apologies for the use of crazy units). This is a total of 23 Million BTU
A single cow contains 750 kg of cow.
Cow has an energy density of just over 10 000 kJ/kg, which is also 10 000 BTU/kg = 7.5 Million BTU/cow
This would mean for each time a Dragon breathes for 32 seconds, they must eat just over 3 entire cows. This is in addition to the regular amount of food they would require.
This is, however, an underestimate. Throw in the fact that the Carnot limit of approximately 70% means that a non magical dragon cannot extract all 7.5 M BTU, and cows would likely be smaller in Essos than with modern farming practices and we could easily round that up to 5-6 cows eaten per 32 seconds of dragon breath.
It's actually a little more plausible than I had first imagined.
I'll leave the aerodynamics maths to someone who knows it better than I, but spoiler alert, it would require a preposterous amount of force for a dragon to fly.