Neither, there were two separate fleets that had access to the Indian Ocean. the first was in Suez, and was stationed there to protect the Red Sea, and most importantly Mecca and Medina, from the Portuguese. A second, and much smaller fleet, was built in Basra, in an attempt to wrest control of the Persian Gulf from the Portuguese. This was ultimately unsuccesful, but the Ottomans were able to use these fleets to protect their territory, and gain some much needed influence in the Indian Ocean.

Source: the only real book I know of about the Ottoman Empire in the Indian Ocean (and the fleets) is by Giancarlo Casale, The Ottoman Age of Exploration

If you are interested in more detailed answer about the Ottomans in the Indian Ocean I would be happy to answer what I can

As a follow-up question, was there any other history of countries or individuals transporting entire ships or other large cargoes across before the Suez Canal was built? Was there a well-established trade/transport route?

From my paper on Islam and Ottoman seapower, I had this as a footnote:

I use “constantly” to mark the fact that such overland transport of ships was, apparently, done on the occasion, as in 1520 when Piri Reis transported a portion of his Mediterranean fleet to Suez by land. Illan Perry, “Migration of Ship-building from the Mediterranean to the Red Sea: Shipwreck of Sharem el Sheikh,” in Ship-building and Navigation in the Indian Ocean Region AD 1400-1800, ed. Satish Chandra (New Delhi: Munshiram Manoharlal Publishers Pvt Ltd, 1997), 110.

Not very many references to it, however - as other respondents noted, the fleets are permanently stationed in the Red Sea and Persian Gulf, so overland movement of the ships were not really necessary once initial presence was established.

What was the structure and size of the Ottoman navy?

Were the Indian and Mediterranean fleets separate completely?

How much control did Istanbul exercise on its subject states such as Barbary states, Morocco, etc?

Any time period is fine.

As a follow up question one of the first battles fought by the Portuguese Empire in the Indian Ocean was fought against the Ottomans, and the wikipedia entry says something like the Ottomans moved their ships through the Suez with technical help of Ragusa and Venice.

Can someone provide me a better source about that (Battle of Diu ?) and about the Portuguese-Ottoman clashes in the Indian Ocean in general?

Hello everyone,

Unfortunately, we have already had to remove a number of poor quality responses in this thread, including many asking about the deleted comments, which merely compound the issue. In this thread, there have been a large number of incorrect, speculative, or otherwise disallowed comments, and as such, they were removed by the mod-team. Please, before you attempt answer the question, keep in mind our rules concerning in-depth and comprehensive responses. Answers that do not meet the standards we ask for will be removed.

Additionally, it is unfair to the OP to further derail this thread with off topic conversation, so if anyone has further questions or concerns, I would ask that they be directed to modmail, or a META thread. Thank you!