The real issue behind your dragon problem isn't biological - it's mechanical. The fastest animal on earth is the Peregrine Falcon, which can only reach speeds of 389 KM/H while going at full-dive, which is several magnitudes slower than needed. Simply put, an organic life form could not possibly achieve supersonic speed...
Unless they can make it into space.
Skydiver Felix Baumgartner have been well-documented to have broken the sound barrier simply by falling high enough - 24 miles up to be exact. If your dragon can get itself that high up in the first place, all it would take is falling down, then catching itself before it breaks every bone in its body.
The problems your dragon would face trying to do this are:
- Getting that high up in the first place
- Surviving the atmospheric climb without externals
- Breaking its fall in a non-catastrophic way
Let's get our dragon into space first.
Getting Up Into Space
If your dragon is Titan-levels of large, and you don't mind it leaving a crater behind when it launches into the air, it could simply push itself off the ground and, with a light enough body, accelerate itself up 24 miles high. This would require an enormous amount of energy, but we're not looking to solve the ecosystem problem of a giant dragon - we just want to get it into space.
Another option, that could pair well with the above solution, would be hydrogen gas pockets that allow it to float up into the atmosphere (and would explain where it gets all that fire, if it can breath fire that is). You wouldn't make it all the way up with just hydrogen, but coupled with a good launch, you might just make it high enough.
This would also have to be aided by wings to give them additional thrust(origin of their wings I will get to later) since any momentum that could solely carry them up that far would create a sonic boom on takeoff (Not that this would be necessarily a bad side-effect).
Getting both of these would be evolutionarily unlikely - you would need a strong food chain full of high-caloric prey, coupled with a reason to float and have such strong legs. Fortunately, I have an idea...
Surviving In Space
Your dragon will need to not only survive in the incredibly low-atmosphere of space, but also survive the lack of oxygen involved with being so high in the first place. Evolution-wise, there's one option that might give you both of these at once: Deep sea hibernating ancestors.
Deep sea ancestors on the scale of Giant Squids would give you giant, strong dragons with especially strong legs that would be increasingly strong outside of the high pressures of the ocean, and hibernating beneath the sea, perhaps after feasting on giant squids, would give them a reason to survive without oxygen.
It even suggests a means by which this behavior was evolutionarily chosen for - launching themselves from the sea floor, the faster and stronger dragons, who could also handle the rapid change in pressure, could more easily catch Giant Squid the faster they can go. Some dragons may have even figured out that they can get a better chance of catching them on the return-trip, which would lead them to trying to jump higher and higher, until eventually they evolve the ability to survive breaching the water, then as they move onto land, learn to use their strong legs to leap into space.
Surviving the Drop
This is your biggest challenge - to explain how a dragon that evolved from a sea creature would survive a 24-mile drop. Their initial sheer resistance to pressure (possibly owing to their scales) helps, but they'll need SOME way to catch themselves as they break the sound barrier.
Fortunately, being from the sea gives them one unique feature that could help them with this - Fins, or evolutionarily speaking, evolved wings. Originally used to slow themselves back down after catching a squid, they could use them to catch the air and shift their direction mid-flight, so that their momentum is reduced and redirected in a wholly non-catastrophic, and entirely sound-barrier-breaking way.
There you have it - dragons that leap into space and come crashing down at supersonic speeds, evolved from giant sea-dwelling predators that ate giant squids literally for breakfast.