Complex sentence: Parts of the birkebeiner were disgruntled by Haakon's policy of reconciliation with the bagler, which may have led to his death, which immediately switched the balance of power within the birkebeiner to the faction around Haakon the Crazy.
Split into simpler sentences: Parts of the birkebeiner were disgruntled by Haakon's policy of reconciliation with the bagler, and it is possible that this may have led to his death. In any case, Haakon's death immediately turned the balance of power within the birkebeiner to the faction around Haakon the Crazy.