Did Jesus and his apostles ever wear turbans?

Yes, they did. Studying the original Greek manuscripts we can see the practice of "wrapping & tying One's Head with materials" is encouraged for important spiritual purposes. This can be seen in the Book of Ephesians. In Ephesians 6 the Word Helmet is used. The original Greek Word here is perikephalaia. Perikephalaia is related to perikeimai, a Greek Word meaning "tie around". Perikephalaia is the original helmet, a helmet constructed of fabrics which are wrapped around ones head in layers bound to the skull, securely fashioned, allowing for quick movements.

It is never directly mentioned in the four Gospels, although it was the mode of dress for Jews as accounted in Exodus 28:3-5: "Tell all the skilled men to whom I have given wisdom in such matters that they are to make garments for Aaron, for his consecration, so he may serve me as priest. These are the garments they are to make: a breastpiece, an ephod, a robe, a woven tunic, a turban and a sash. They are to make these sacred garments for your brother Aaron and his sons, so they may serve me as priests. Have them use gold, and blue, purple and scarlet yarn, and fine linen."

For further references look up:

Job 29:13-15

Zechariah 3:4-10

Revelation 19:12