Morgoth's curse on the children of Hurin was effective without the action of Glaurang. I believe the curse was at least partly at work with Turin fleeing from Doriath after the accidental slaying of Saeros, the killing of his friend Beleg Strongbow when he was freeing Turin from the orcs that had captured him. Had none of these things happened, Turin would probably have never gone to Nargothrond, where his actions led to it being discovered by Morgoth and Glaurang and its ultimate destruction, and the events that followed later involving Nienor.
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His sword was long his lance was keen
His shining helm afar was seen
The countless stars of heavens field
Were mirrored in his silver shield
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