Don't amp turns found in a magnetic element depend nearly entirely on current?
Yes and No.
In the steady state, the magnetic field is directly related to the amp-turns.
But creating the magnetic field needs energy. As you try to put more current, there is a back-emf produced and that voltage must be overcome to put more current.
In an inductor, the energy is stored into the magnetic field.
For a capacitor, the energy is stored into the electric field. You need to spend energy to increase the electric field from 0 to E.
In the same way, you need energy to create a magnetic field from 0 to H.
Energy is associated with the field, magnetic of electric. For a resistor, the energy is dissipative (joule heat; that cannot be recovered).
You can recover the energy from the magnetic field (or the electric field) when the field collapses.
You need to spend energy in a superconductor to get to a high current (see
https://en.wikipedia.org/wiki/Type-I_superconductor for some more).
When the superconductor is quenched, all the energy is dissipated as heat.