The English words turban and tulip are doublets. Doublets (or etymological twins) are words in one given language that go back to the same etymological source but differ in form and meaning—cf. also lobster – locust, fawn – fetus, pastiche – pastis and clock – cloak.

 

The word turban is from tul(i)pant, a vulgar-Turkish pronunciation of Persian dulbănd or dōlbănd, meaning turban.

This is also the origin Italian, Spanish and Portuguese turbante, French turban, Dutch tulband, German Turban, Danish and Swedish turban.

In English, the word is first attested as tolipane in A compendios and briefe declaration of the iourney of M. Anth. Ienkinson, from the famous citie of London into the land of Persia, passing in this same iourney thorow Russia, Moscouia, and Mare Caspium, aliâs Hircanum, sent and imployed therein by the right worshipfull Societie of the Merchants Aduenturers, for discouerie of Lands, Islands, &c. Being begun the foureteenth day of May, Anno 1561, and in the third yere of the reigne of the Queenes Maiestie that now is: this present declaration being directed and written to the foresayd Societie (1561), in which the British traveller Anthony Jenkinson (1529-1611?) thus described Obdolowcan (i.e. Abdullah Khan), the governor of Shirvan, in the eastern Caucasus, a tributary of the Safavid empire, which ruled Persia:

A prince of a meane stature, and of a fierce countenance, richly apparelled with long garments of silke, and cloth of gold, imbrodred with pearles and stone: vpon his head was a tolipane with a sharpe end standing vpwards halfe a yard long, of rich cloth of golde, wrapped about with a piece of India silke of twentie yards long, wrought with golde, and on the left side of his tolipane stood a plume of fethers, set in a trunke of golde richly inameled, and set with precious stones: his earerings had pendants of golde a handfull long, with two great rubies of great value, set in the ends thereof.

 

The English word tulip, formerly tulipa, tulpia, tulipant, etc., has the same origin, because the expanded flower resembles a turban. In The Herball or Generall Historie of Plantes (1597), the herbalist John Gerard (circa 1545-1612) explained:

After it [= the Tulipa of Bolonia] hath beene some fewe daies flowred, the points and brims of the flower turne backward, like a Dalmatian or Turkes cap, called Tulipan, Tolepan, Turban, and Turfan, whereof it tooke his name.
[…]
The later Herbaristes by a Turkish and strange name call it Tulipa of the Dalmatian Cap called Tulipa, the forme whereof the flower when it is open seemeth to represent.
It is called in English after the Turkish name Tulipa, or it may be called Dalmatian Cap, or the Turkes Cap […] In the Turkie toong it is called Café lalé, Cauále lalé, and likewise Turban and Turfan, of the Turkes Cap so called.

The first mention of the tulip by a Western European is in a letter dated September 1554 but written in 1555 by Augier Ghislain de Busbecq (1522-92), a herbalist and diplomat who served as Austrian ambassador to the Ottoman Empire. He wrote that, on the way from Adrianople to Constantinople, he and his companions

everywhere came across quantities of flowers—narcissi, hyacinths, and tulipans, as the Turks call them. We were surprised to find them flowering in midwinter, scarcely a favourable season. There is an abundance of narcissi and hyacinths in Greece, and they possess so wonderful a scent that a large quantity of them causes a headache in those who are unaccustomed to such an odour. The tulip  has little or no scent, but it is admired for its beauty and the variety of its colours.

In the original modern-Latin text, published in Itinera Constantinopolitanum et Amasianum in 1581, “eorum quos Turcae tulipan vocant” literally means “those that the Turks call tulipan” and “tulipanti aut nullus aut exiguus est odor” “the scent of the tulipan is either non-existent or faint”.

The Turkish word is also the origin of French tulipe, Italian tulipano, Spanish tulipán, Portuguese túlipa, German Tulpe, Dutch tulp, Danish tulipan, Swedish tulpan.

The tulip is called lale in Turkish and لاله (lāla) in Persian.