I don't think "fresh water" ice is made from salt water. The glaciers, which are the source of most of the icebergs and ice floes, are made from snow that is compacted down into ice over a long period of low temperatures and additional snowfall on top. The evaporation of salt water produces fresh water, which then falls as rain or snow.

When water freezes, all the individual water molecules (which are roughly "V" shaped) click together into a highly structured repeating 3D crystal pattern that looks like this. Note all the waters form hexagons, kind of like a honeycomb - that's why snowflakes are hexagonal and have 6-sided symmetry.

So yes the water does "purify" itself as it crystallizes into solid ice. Basically, the salt molecules don't really fit or snap into this hexagonal pattern very well, so they mostly get left behind as all the waters start clicking together into their big solid scaffold.

Ice floats because water gets less dense when it freezes. But that isn't true for salt water - salt water behaves like most other substances and increases in density as it freezes. This means that when the ocean gets near freezing, cold, salty water sinks leaving warmer, less salty water near the surface.

Sea ice will only form when the temperature has dropped to a few degrees below freezing for an extended period of time because that allows for the salinity at the surface to drop to the point where ice with that level of salinity has slightly positive buoyancy (it floats). The level of salinity where ice will float is similar to the level of salinity in brackish waters in the ocean near the outflows of major rivers. Ice with that level of salinity is still too salty to be called fresh water, even though it is less salty than the ocean.

But there's another process that occurs - because salt makes it harder for water to freeze, the water that freezes first is formed of microscopic pockets of water that just happens to be salt free. Those initial, salt free ice crystals grow by "absorbing" water molecules from the surrounding ocean. IE, molecules of salt won't stick to those initial ice crystals, but other molecules of water will.

As the salt free ice crystals continue growing, they begin to trap microscopic pockets of very high salinity water inside of them. As a result, new sea ice is made up of crystals of pure, salt free ice water with millions of tiny pockets of extremely salty water trapped inside. Given how small the salt water pockets are, the ice will still appear to have the same brackish salinity as the sea water it froze from to something the size of a human.

But those pockets of salt water don't remain static. The salt will "melt" through the fresh ice below it, eventually melting all the way back into the ocean. This leaves behind a tiny pocket of water free air. The longer a piece of sea ice exists, the more opportunity the salt water inside of it has had to melt out. After about a year, all of the salt inside of the ice will have melted out and what you're left with is a hunk of pure frozen water.

Fun fact, those pockets of ice are what get brinicles rolling.

This is why salt, and other dissolved compounds, will reduce the freezing temperature of water: the interfere with the formation of the crystal lattice structure required for the phase transition

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And when sea ice is forming, underneath the ice layer it forms a brinicle of highly salty and colder water.

https://www.youtube.com/watch?v=BtQhb8sWJNw