It's hard to put a precise date without generating debate, but I'd put forward that Europe took the lead at some point between the early 14th to late 15th century.
Several processes were in full motion around then:
Where there was indeed a dearth of books and literate scholars in the Early Middle Ages, science and technology in the Mid- to Late Medieval Europe actually was a lot more dynamic than is often credited. It saw developments such as the Three Field System, plowing-related technologies, mill-related technologies, etc. - all of which contributed to significantly boost the agricultural output.
Also worth highlighting during the period are universities, which began to appear between the 11th and 13th century depending on the region.
My point here is that it's incorrect to assume Europe was roamed by uneducated peasants only. And the Byzantine Empire to the East was far from backwards, and still interacting with Western Europe. Science never really died during the period.
Fibonacci is known nowadays mostly for the sequence named after him, but the latter actually is a mere example problem he gave in passing within Liber Abaci, a book he published in 1202.
Liber Abaci introduced Hindu-Arabic numerals and place value to Europe, enjoyed a very wide success. It was transformative for both science and finance in Northern Italy and then Europe.
Being a savvy marketer in his time, he followed it with a shorter, more to the point version for use by merchants. One could argue that it laid the foundations that were necessary to lift Europe out of the middle ages.
When one thinks Renaissance, one usually thinks about the rediscovery of Classic arts and literature. But this is actually hiding the reality that preceded it in the late Middle Ages.
Numerous scholars went to Spain to study the very large collection of scientific literature that the Muslims had left behind - particularly after Seville fell in 1248. (Sicily arguably was a major cultural gateway before that.)
Gutenberg's invention of metal movable type around 1440 allowed to mass print books.
The bible was, of course, the top seller. Most titles below it were scientific in nature. (In particular works based on Fibonacci's book.)
Waves of Byzantine scholars and émigrés came to Italy in the aftermath of the Turks conquering Constantinople and dismantling the Byzantium Empire in 1453.
They brought numerous manuscripts with them, ultimately setting the stage for the Renaissance.
Depending on the historian you ask, the pivotal point occurred between the 14th and the 16th century. Whichever date it is, what's clear is that the period saw the "rise of a clerical faction which froze this same science and withered its progress".
The Tarqi ad-Din observatory in Istanbul, for instance, was ultimately destroyed by a squad of Janissaries "by order of the sultan, on the recommendation of the Chief Mufti" around 1577.