If you compress water strong enough will it turn into ice?

at what temperature? at room temp, 25ºC, yes, if the pressure is
greater than 1 GPa (10000 bar). That would be ice VII (ice has many different crystalline structures)

but it would go back to liquid as soon as the pressure is released.

it would change to vapor (boil) if the pressure dropped below 3 kPa (normal 1 atm pressure is 100 kPa)

Yes, it just depends on the amount of pressure.  Here is one example.
https://en.wikipedia.org/wiki/Ice_VII

To add to Bill Russel's answer. At high pressures, but less than the pressure described, ice skates can cause a small part of the ice to melt. Giving a lubricating film of water. This is different from most other materials.

The answer is an unqualified "yes."  If you look at Bill Russell's nice phase diagram, you'll see that even at 400C, a pressure of a few hundred thousand atmospheres would suffice to compress the water to ice.  That pressure may sound high, but consider that the pressure in the core of the sun is around 250 billion atmopheres.  What actually happens in an environment like that is that any water would be pushed out in favor of substances with higher densities.  Devices on earth have been constructed to exert pressures of several million atmospheres, so could compress fairly hot water to ice.

you have to go to very high pressures before the atypical negative P/T slope of the solid-liquid phase boundary reverses and you can make ice by pressurization rather than the reverse as happens at lower pressures (pressurization can cause melting at low temperatures).

That is, even though increasing pressure makes liquid water more stable (stable over a wider range of temperatures) at low pressure, eventually that does reverse and water/ice behaves as all other solids do, where pressure makes melting more difficult rather than easier.  We don't see any such pressures in our near earth-surface environment, so our practical experience is that pressure on ice can lead to melting rather than solidification from liquid as is generally the case.

Eventually, though, the pressure dominance does take hold.

Sure, it will rise the freezing point temperature up !!!

Theoretically if you compress water enough you lower it's freezing point and water can become ice at room temperature

No, when you compress it is absorbs heat.
If you put water into a vacuum chamber it will boil until it freezes.