How Secure Are the US H-Bombs in Turkey?

Not very, apparently. The US had power cut to the base and a no-fly zone including the base was called.

I don't think that would deter them from taking off if they really wanted to but that would strain US/Turkish relations even further than they are already strained at the moment, which is just before the breaking point.

There have been quite a few calls in the last couple of months to remove the weapons from Turkish soil due to perceived (and apparently correctly perceived) instability and an accelerating slide into a dictatorship.