Túrin was never cursed
The idea that Morgoth can curse the lives of Men, who explicitly are not tied to fate, is ridiculous. The whole 'curse' is something Morgoth made up to mess with Húrin- Túrin's misfortunes were entirely due to his own actions, and whenever it seemed like his life was about to stop sucking, Morgoth sent Glaurung or his orcs out to screw it up again. Even things like Gwindor escaping could have been part of his plan; by informing Túrin of the 'curse', he became more desperate to struggle against it, ironically sealing his fate.
- Tolkien himself leaves this ambiguous- Húrin himself believes that such a curse is beyond Morgoth's power, but Morgoth himself is absolutely confident that he can, or he wouldn't have made Húrin watch (though admittedly, Morgoth isn't really the best judge of his own capabilities, being a megalomaniac and all, and as Morgoth he's far weaker than he was as Melkor on top of that). Myself, I'm of the opinion that it was a bit of both- Morgoth's thought did shadow Túrin and Niënor, but wouldn't have had nearly the effect it did if it weren't for their own natural penchant for getting themselves in trouble (to say nothing of their misfortune in encountering Glaurung, his then-current- and literal- Dragon).
- During Turin's stay in Doriath Morgoth got actually frightened, thinking that maybe Turin would manage break the curse after all. That supports the theory of Morgoth could manipulate events and plant seeds of future disgraces... but he could not control someone else's fate. Turin's pride, inability to listen to good advice and be humble and poor choice-making skills drove him to fall into every Morgoth's trap.