Stanley Wells' edition of A Midsummer Night's Dream (The New Penguin Shakespeare, 1967, 1995) has the following gloss for "Tartar":
The Oriental bow was of special power. The image may have come to Shakespeare by way of Golding's translation of Ovid's Metamorphoses, X.686-7, 'she / Did fly as swift as arrow from a Turkey bow'.
R. A. Foakes' edition of A Midsummer Night's Dream (The New Cambridge Shakespeare) has the following gloss for "Tartar's bow":
Noted for its power; the image was probably derived from Golding's Ovid, Metamorphoses X, 687, where Atalanta runs 'as swift as arrow from a Turkye bowe.' The word 'Tartar' was used vaguely to signify anyone from central Asia, including Turkey.
Sukanta Chaudhuri's edition of the play (The Arden Shakespeare, third series, 2017), essentially agrees with the above and adds a few more pieces of information:
Shakespeare was probably echoing Golding 10.687, 'as swift as arrow from a Turkye bowe', where Oved (Met. 10.588) has 'Scythica ... sagitta', 'Scythian bow'. Ancient Scythia overlapped with later 'Tartary', and Turk and Tartar were popularly associated: see 263, and OED Tartar n.² 1. The Turks had a venerable tradition of archery, reaching its highest point during the Ottoman Empire, and known to Europeans from the time of the Crusades. 'As swift as an arrow' is proverbial by itself (Dent, A322).
('Dent' refers to Shakespeare's Proverbial Language: An Index by R. W. Dent (1981).)
Based on this, it is not the bow that is fast, but arrows shot from that type of bow (which sounds more plausible at any rate).
Ovid's Roman contemporaries knew archers on foot (the sagittarii) but armies that relied a lot on mounted archers were able to inflict major defeats on Roman armies, for example at the Battle of Carrhae in 53 BC, where the Parthians defeated a numerically superior Roman army led by Crassus. This type of defeat led to the introduction of mounted archers in the Roman armies during the Principate.
The Scythians, the Parthians and the Turks all used composite bows, whereas Europeans, including the English, used self bows (i.e. made from a single piece of wood).
It is also worth knowing that
Turkish archers developed several unique techniques to aid in combat. One was the practice of holding several arrows in between the fingers of the draw hand, allowing fast repeat shots.
This could mean that Turkish archers may have been able to fire more arrows per minute than their English counterparts.
Mounted archers seem to have relied more on a thumb draw, i.e. using the thumb instead of the index and/or middle fingers to draw the string. For a description and explanation of bow types, arrow types and techniques used by eastern mounted archers, see the video Firing Arrows Like a Mongolian Warrior on The Modern Rogue (YouTube). (This video also does away with the misconception that a longer bow is necessarily more powerful than a shorter composite bow.)
With regard to the question what Elizabethans could know about the Turks, it is also worth noting that Elizabethan England did have some sort of diplomatic relationship with the Ottoman Empire. When Elizabeth I ascended the throne, she became the head of a non-Catholic nation that was surrounded by Catholic adversaries (Ireland, France and above all Spain). Using the dubious logic of "the enemy of my enemy is my friend", Elizabeth sent emissaries to the Shah of Iran, the king of Morocco and the Ottoman Sultan Murad III. In 1599, she even sent a self-playing organ as a present to Sultan Mehmet III of Turkey. One of the craftsmen that went to Turkey with the organ was Thomas Dallam, whose diary forms the basis of the book The Sultan's Organ (translated by John Mole and published in 2012).