Pilot on functional check flight reported engine malfunction inflight. Pilot diverted and landed uneventfully.

Narrative

After departing on an FCF (Functional Check Flight) in Aircraft X the left engine started stuttering. I checked the engine instruments and noticed the LH TIT was in the red arc and the EGT was high; occasionally bouncing into the red. None of the indications I had led me to believe the engine was going into cylinder deactivation (no overspeed or over boost). I advised ATC we would be diverting; and told them the reason was the engine wasn't performing the way we wanted it to for the maintenance test; but we did not need any assistance at the time. ATC began vectoring us toward ZZZ and I advised the company we were diverting. I told departure we were ready to go into ZZZ and they gave us a descent to 2;500 ft. The mechanic on board then pointed out a flame coming from the LH exhaust. I looked out and saw a large blue flame constantly coming from the exhaust pipe. I tried reducing the throttles and RPM to see if it helped and it didn't. After cross checking instruments; which still showed high TIT and EGT; along with the continuous flame coming from the exhaust; I decided the best course of action was to shut down the left engine in accordance with the QRH. I told ATC I would need delay vectors; that I would be shutting down an engine. After running through the immediate action items I gave the mechanic the QRH so he could read me the Engine Failure; Fire or Severe Mechanical Damage" checklist while I was flying. We successfully secured the engine and completed all applicable procedures. We then continued inbound to ZZZ where we landed safely. Based on the engine instruments I observed during the event; I believe the turbocharger overheated. Oil PSI and TEMP; as well as fuel PSI were all within normal ranges. When the engine was skipping; the MAP would drop momentarily but it didn't immediately concern me because it never remained low for more than a moment. The only indications that gave me an idea of what was happening were the TIT and EGT; which are both related to the turbocharger temperature. That; coupled with the exhaust flames; give me the impression the turbocharger was overheating."