A capacitor can deliver a large current for a short time. You don't say how much voltage you need to create the 50 A, but let's say 1 V. That means your wire is 20 mΩ, which is doable. Let's say 2 V to account for other drops in the system.

Let's further say that you want the current not to deviate more than 20% at any one time from the 50 A average. This will allow the current to start at 60 A and drop to 40 A during your 20 ms test time. From that, just do the math:

(50 A)(20 ms) / (20 V) = 50 mF

That's only five 10 mF capacitors in parallel, for example. Make sure they can produce the short term current. "Super caps" generally can't do that. A few decent size electrolytic caps should do it. Note that you don't need high voltage.

A capacitor can deliver a large current for a short time. You don't say how much voltage you need to create the 50 A, but let's say 1 V. That means your wire is 20 mΩ, which is doable. Let's say 2 V to account for other drops in the system.

Let's further say that you want the current not to deviate more than 20% at any one time from the 50 A average. This will allow the current to start at 60 A and drop to 40 A during your 20 ms test time. That would mean 2.4 V down to 1.6 V, for a drop of 800 mV. From that, just do the math:

(50 A)(20 ms) / (800 mV) = 1.25 F

That's one whopping big capacitor, or more likely, quite a bank of capacitors in parallel, although they only need to go up to 2 V.

This shows that more turns of thinner wire makes the problem easier since you can use a higher voltage. A car battery can produce 50 A for well more than 20 ms. You say you only have room for 4 turns of wire, but with thinner wire you can fit more turns. At only 20 ms duration, you don't have to worry about the wire vaporizing.

Try maybe some #22 magnet wire to see how many turns you can pack in there. That will also have more resistance, but require less current. Both those effects will help in getting a voltage source that can power it for the bursts you need.