This usually means air in the line rather than anything suspended in the water.

Pressure drops as the water leaves the tap and dissolved air comes out as tiny bubbles, which rise, so the glass clears from the bottom up as they go.

Look for a glass that clears within a minute from the bottom upward, white or milky rather than colored cloudiness, and nothing left in the bottom of the glass.

Do not buy a filter for this. Air is usually not a filter problem, and a cartridge will not change what you are seeing.