[Drewbert]

Ah, but as the ice melts faster, channels of water start forming below the glaciers and instead of ice melt being the biggest cause of the rising water level, you get huge mountains of ice suddenly sliding into the sea.

Huge mountains of previously landbound ice going into the sea equals instant rapid sea level increase. Once the ice is in the sea, the melt rate matters not, as future melting does not affect the sea level.

Try it with a glass and some ice cubes.

They've already detected a significant increase in water flows underneath the ice in Greenland.