You should use a current transformer. That can be AC which is just like a conventional transformer or a Hall Effect one that will work at DC
You cannot just wind a few turns around the conductor. That is going in the wrong direction. The magnetic field goes around the conductor, so to induce a current the wires should be parallel to the main conductor.
The most usual form of a current transformer is a toroidal with a single turn primary. You could use any toroidal as a current transformer as long as you can fit the primary down the middle.
You must put a suitable load on at least one of the windings or there will be big voltages.
If you use a toroidal transformer of about 30 VA, it will have about 0.25 V per turn, so on 120 V primary that is about 480 turns, and on a 6 V secondary that is about 24 turns.
If you have a single 30 A turn as your new primary, that will give you about 1.166 A on the 6 V winding or 20 mA on the 120 V winding. You shouldn't exceed the rated voltages as a load, so either 5 Ω on the 6 V winding or 1920 ohms on the 120 V winding. However, as all you want to do is sense the current, a 250 Ω load on the 120 V winding will give you a 5 V signal.
There will be a voltage drop, but it will be tiny. If the secondary voltage is 5 V, the primary will be about 10 mV for that sort of turns ratio.