Based on your description it sounds like you have an exhaust manifold that's leaking between the head and the manifold or between the manifold and the turbo. Both areas have their own gaskets to seal the connection. I don't think it's too common on TDI's but was fairly common on the Audi 5cyl's where the Exhaust Manifold would warp enough over time to leave a leak between the head and the manifold. Putting a new gasket on it might solve the problem only short term, the solution would be to have a machine shop take a small cut off the mating surface. Just hold up the exhaust manifold to a straight edge before reinstalling it with new gaskets, that way you'll know the manifold isn't the culprit and you won't have to do the job over again.
It's gonna be kind of a pain in the butt to take all that apart for a couple reasons, you'll have to take off the turbo, drop the downpipe, intake manifold too. All the while you'll be bent over the engine reaching back to the work area for extended times. Probably the least comfortable vehicle working position in my opinion.
Good luck, let us know what you find.