Abarim Publications' online Biblical Hebrew Dictionary
It's not clear whether there are two separate roots of the form טבל (tbl) or only one:
The verb טבל (tabal I) expresses a whole or partial submerging in liquid: the dipping of a tunic in blood (Genesis 37:31), a finger in blood (Leviticus 4:6), hyssop in water (Numbers 19:18), a foot in oil (Deuteronomy 33:24), a staff in honey (1 Samuel 14:27), bread in vinegar (Ruth 2:14), a person in a pit (Job 9:31), and a Aramaic general himself in the river Jordan (2 Kings 5:14).
If there ever was a verb טבל (tabal II), it doesn't exist in the Bible, and its existence is proposed solely to explain the occurrence of the masculine noun טבול (tebul), which denotes a turban (Ezekiel 23:15 only).
BDB Theological Dictionary makes mention of an Ethiopian verb that means to wind about or wrap up and that may be comparable to our illusive verb טבל (tabal II), but also notes the suggestion of other scholars who have proposed relations between our noun and an Arabic noun meaning dye, which in turn could be linked to the verb טבל (tabal I).
Whatever the true origin of the noun טבול (tebul), to the Hebrews a turban was either something to "dip" one's head in, or else, a garment with which one could advertise one's status or place or origin and that by means of its color or dye.
Turbans were worn by many peoples (including Jews) and it's not wholly clear what might have prompted it. Since in as many cultures submersion in water grew from a practical act of cleaning to a symbolic act of purification, it might be considered that to the Hebrews the wearing of a turban signified purity (in a cultural status sort of way).