Hi,
It's not *only* that it collapses, it's *also* how fast it collapses (or how fast it changes).
v=K*d(Phi)/dt
In the above, d(Phi)/dt is the rate of change of flux. Obviously the larger this is the larger the voltage will be with any constant K, and K could be the number of turns of the coil.
The division d(Phi) divided by dt could be looked at as an increment in the flux divided by the increment in time that it took to see that said increment in flux. If we see a larger change in flux for a given amount of time, that means the rate is higher, and so multiplying by the constant K gives a larger value of voltage v.
The value K multiplies the rate, so 50 turns would produce 5 times the voltage as 10 turns would.
So there are two ways to get a higher voltage:
1. Increase the number of turns.
2. Increase the rate of change of the flux.
Sometimes you can not change either of these things because they are already set by the design of the coil and external parts.