send them a letter, ask them to put their denial on paper, state that you require them to explain how the aftermarket exhaust caused this failure, ask them to respond in 10 days. Send it certified return receipt mail. Address to the service manager or dealership management.
Depends on what we are talking about when we say "exhaust", you mean a DP that allows the turbo to spool up faster and higher? You mean a fart-can axle-back that does nothing but change the noise? There's some in-between with cat eliminations and the such, but typically if you are retaining O2 sensors and that kind of stuff and not re-tuning, it's generally not putting any additional stress on the turbo.
The exhaust in my WRX, oh heck yea, I was running a large DP to larger exhaust which with the tune made the car spool the turbo up to a higher PSI than stock, for sure that put more stress on the parts. But then fart-can stuff that most refer to as "exhaust", that can't change the amount of wear and tear on the turbo.
You can't screw around with calling people or talking in person, they can deny that ever happened unless you get real aggressive subpoenaing ridiculous stuff like phone logs and surveillance cams, which is unlikely to happen. But getting this on paper, most businesses will "do the right thing" when you start serving them with paper, because they KNOW that is a record that can and will be used if the party takes them to court.
Plus, the amount of cost for this is probably such that you could take it to small claims court with no trouble at a minimal expense. This is where your written record is gold. Keep all records. Write summaries of each voice conversation you have had. Get the names of the people you have talked to. IF they are being evasive or won't supply the names, write that stuff down too.