As has been mentioned in other answers, the earliest meaning of turnpike offered by the OED refers to a barricade "as a defence against sudden attack, esp. of men on horseback."
Several of the citations provided seem to suggest that a turnpike in this sense could be opened or closed, which explains the use of the word "turn."
My lord hath do brokyn all the passages excep Newhom bryge, weche ys wached and the turne-pyke shette euery nyght.
- 1477 E. Bedyngfeld in Paston Lett. & Papers (2004) II. 420
They had no Drawbridge but only a Turnpyke.
- 1644 in J. Rushworth Hist. Coll.: Third Pt. (1692) II. 739
From the meaning referring to a barrier that can be opened or closed, it makes sense that later meanings referred a toll barrier or a turnstile, whether on roads or waterways.
†2. A horizontal cross of timber turning on a vertical pin, set up to exclude horse-traffic from a foot-way: a turnstile. Obs. [Attested 1545]
†3. A barrier across a watercourse or stream; a water-gate, allowing the water to flow, but obstructing cattle; also, a lock on a navigable stream. Also turnpike-lock (see Compounds 1). Obs. [Attested 1623]
4.a. A barrier (orig. of the nature of a turnpike in sense 2, later a gate or gates) placed across a road to stop passage till the toll is paid; a toll-gate. Cf. turnstile n. Now chiefly Hist. [Attested 1695]
Finally, attested in 1748 is a definition that refers to it simply as a shortening of "turnpike road," which has its own separate entry in the OED.
A road on which turnpikes are or were erected for the collection of tolls; hence, a main road or highway, formerly maintained by a toll levied on cattle and wheeled vehicles. Also fig.
The first attestation for this meaning is provided only shortly before the attestation of the ellipsis to simply turnpike:
Turnpike roads were not known in that part of England till some years after.
- 1745 J. Wesley Wks. (1830) I. 485
So it seems from examining the entries in the OED that a turnpike originally referred to a barrier, likely a movable one that was meant to block intruders, and this term was adopted into the compound form turnpike road, meaning a road that contains turnstiles for paying a toll, which was in turn shortened back to turnpike, hence the meaning referring to roads with toll passages commonly used in the U.S.