Thanks for the pictures.
Please take that PC off that comforter ASAP. Your power supply isn't getting ample airflow. The air intake is on the bottom of the case.
Something that is strange is that it works with your R9 380, but not your GTX 1060; the 380 has a higher power usage. These issues usually are related to heat, power, or software. Honestly I would agree with you that your power supply a good candidate for the cause of the problem except the 1060 has a lower power consumption.
Addendum: What do you mean by the 1060 steeps down? The weight of it is drooping the PCIe slot? If that's the true, could you place the computer case on it's side to test? Meaning, place the open computer case, so that the motherboard is facing the ceiling. I suggest leaving the case open, so that excess heat can escape, while we're testing.