Did the Ottoman empire suppress the printing press?

There is some truth in it, but the printing press adoption delay was only a late symptom of an attitude that began much earlier. In short is was more of contempt for the culture and aesthetics of print and the demand for censorship than an outright real, complete ban.

The first book in Arabic was published in an Italian town named Fano in 1514. During those times, books in Arabic and Farsi were also published in Venice, Rome and Vienna. These were usually books that merchants and missionaries could make use of. There were printing houses in Istanbul called "basmahane" in the 15th century. However, these printing houses were managed by Greek, Armenian and the Jewish community and they printed books in their own languages. In 1493, there was a Jewish printing house in Thessaloniki. Armenians learned the printing press business in Italy and opened a printing house in Istanbul in 1567, followed by the first Greek printing house in 1627. The very first thing that the Greek printing house printed was a booklet targeting the Jews. Sultan Murad III signed a firman, an official decree, in 1587 allowing the sale of the books in Arabic, Farsi and Turkish, which were published in Europe, on Ottoman soil.

Yet, the Turks did not like books that were printed in the printing houses, but rather preferred hand-written ones. The published books lacked the art and grace of hand-written books. Ottoman intellectuals, who were keen on aesthetics, enjoyed books written with elegant handwriting and whose ink shined, along with edges that were ornamented with golden gilt and covers that were made with care. Reading books was not only a necessity, but also a pleasure. Besides, there were many calligraphy artists who copied plenty of books rapidly. All these people could be out of a job. On top of that, those who were keen on books belonged to a certain class, just as today.

The first printing press, which belonged to the Muslims, was established by Ibrahim Müteferrika in 1727, during the Tulip Era when the Ottoman industrial revolution began. It was opened 36 years later than the first printing press in New York.
_{–– Ekrem Bugra Ekinci: "Myths and reality about the printing press in the Ottoman Empire", Daily Sabah: Istanbul, 08.06.2015.}

And that's not just the modern Turkish perspective on the subject, albeit a biased account it is not entirely freely invented nonsense. More free-floating accounts exist.

After a real flourishing of science and technology during almost the entire Middle Ages, that general but not universal decline started in the 12th century and began to accelerate during the 14th and 16th century.

But just the printing press as a quite simple to copy tech:

The invention of printing by the unbelievers is a very strange art, and verily an unusual invention . . . . [I]t was devised in the year 1440 in [Mainz] by a wise man called Aywan Kutanbark [i.e., Johannes Gutenberg] . . . . [S]ince then all the books by the unbelievers are produced by printing . . . . When one intends to print a book it is as hard as handwriting to arrange the types in lines. But once arranged one thousand copies can be printed in less time than copying one volume by hand.

extant documentary evidence does not support the claim that Ottoman sultans banned print. What it does support, however, is scholars’ longstanding attempt to explain the Ottoman experience of printing through that of Europe. Ban or no ban, it is this perspective that ought to be made an object of study and finally dislodged from the foundational core of scholarship on Ottoman printing.
_{–– Kathryn A. Schwartz: "Did Ottoman Sultans Ban Print?", Book History, Volume 20, 2017, pp. 1-39. DOI.}

For the printing press:

A fundamental puzzle of technological history is why some societies have foregone free lunches by failing to adopt technological advancements completely (Olson, 1982; Mokyr, 1990). In one of the best-known and most puzzling cases of foregone opportunity, it took the Ottomans nearly three centuries after the invention of the moveable type to sanction and offer explicit support for printing in Ottoman Turkish (in Arabic characters) in Istanbul in 1729. The delay has led to numerous speculations about Muslim reaction to new technologies, inviting various types of explanations. Some historians have attributed the delay to cultural values such as religious conservatism and obscurantist thought, others looking for the answer in socio-economic factors such as entrenched interests and institutional rigidity.

For a satisfactory explanation of when a society adopts a new technology and why there may have been delays and restrictions, we need to identify not just the factors that may have obstructed change in some technologies but also those that have facilitated the swift adoption of technological advancements in other areas. The case of the printing press becomes even more puzzling when we consider it in relation to other technologies that were adopted quickly during the same time period. Contrary to the image of the religious and technological conservatism that seems consistent with the delayed adoption of the printing press, the Ottomans were eager to adopt the latest advancements in military technology such as the use of gunpowder and firearms (Agoston, 2005). In adopting the printing press and gunpowder weapons, often considered the most important inventions of the late Middle Ages, the Ottomans reacted quite differently, displaying a mixed image between conservatism and openness and making it difficult to explain their reaction through ad hoc factors.

The effect of technology on legitimizing relationships explains the differential reaction of the Ottomans to advancements in printing and military technologies. They regulated printing technology heavily to ensure that it did not decrease the ruler’s net revenue by undermining the legitimacy provided by religious authorities. But they readily accepted new military technologies such as gunpowder and firearms because they increased the net revenue the ruler could collect from the citizenry while having a positive effect on the military’s ability to legitimize. Our approach also explains why the Europeans were quicker to accept the printing press and why the Ottomans eventually adopted it. Heavy regulations did not last as long in Europe as they did in the Ottoman Empire because different, non-religious sources of legitimacy had already emerged. Although religious legitimacy was still important to European rulers, other important sources of legitimacy were available by the advent of the press. In the same way, the Ottomans sanctioned printing in Arabic characters in the eighteenth century only after alternative sources of legitimacy emerged.

The Ottoman reaction to the printing press was different from their reaction to military technologies and the reaction of the Europeans to printing press. Although mass printing could have raised economic productivity and the size of the surplus available to the ruler for taxation, the Ottomans chose to forego the opportunity and regulate the technology heavily for almost three centuries by banning printing in Arabic characters. They were not initially enthusiastic about the new technology because they expected it to lower their net revenue by undermining the ability of religious authorities to generate loyalty. The European rulers, on the other hand, were generally open to the adoption of mass printing because they relied on religious legitimacy to a lesser extent at the time of the invention of the press.

The Ottomans started to relax the restrictions on the printing press in the eighteenth century. New sources of legitimacy gained importance in the intervening centuries, and hence it mattered less that the printing press threatened the ability of religious authorities to produce loyalty. Its expected benefits to the ruler’s revenue had also increased, so the Ottomans deregulated the technology when its expected benefits exceeded the cost.
_{–– Metin M. Coşgel & Thomas J. Miceli &Jared Rubin: "The Political Economy of Mass Printing: Legitimacy and Technological Change in the Ottoman Empire", Journal of Comparative Economics 40.3 (2012): 357-371. DOI: 10.1016/j.jce.2012.01.002}