Tyr had unknowingly broken a rule imposed by this clan, he had just executed a person he 'deemed' guilty without a trial and even Kolbeinn had a look of disappointment on his face because there was nothing he could do.
Tyr's story was not believed, because Tyr could have very well beaten Finn and killed the other man when he tried to stop him.
Then try to have his way with Nora but Nora was too scared to speak up against her Chieftain so this was the ultimate result.
Kolbeinn knew he had to look at every possible angle, he could not be biased just because it was a son of Ragnar that committed this crime.
It would have made more sense if Tyr kept him alive for him to confess to his sins and not take matters into his own hands.
The dark-skinned boy was forced on his knees in front of Kolbeinn, Kolbeinn looked at Tyr.