Dear student,
field is not zero inside when there is a steady current flow inside a conductor like copper.
ELECTROSTATIC FIELD inside aconductor is zero, not electric field. ELECTROSTATIC field means the electric field created by charges at rest.
We can understand this property by considering a conducting slab placed in an external field E. We can argue that the electric field inside the conductor must be zero under the assumption that we have electrostatic equilibrium. If the field were not zero, free charges in the conductor would accelerate under the action of the field. This motion of electrons, however, would mean that the conductor is not in electrostatic equilibrium. Thus, the existence of electrostatic equilibrium is consistent only with a zero field in the conductor.
Thus when charges are in motion, it means that there is no ELECTROSTATIC equillibrium, thus ELECTROSTATIC field is nonzero.
Now if we apply a potential difference across a conductor, we have created an electric field inside the conductor. Thus electrostatic equillibrium is disturbed and electrons flow under applied filed, and so does the current.
I hope you understand.