Museum of Calligraphy (Istanbul Turk Hat Sanatlari Muzesi) is the only museum dedicated to Calligraphy in Turkey. It is located in the west side of Beyazit Square in a building that is a part of the medrese of Beyazit complex. Unfortunately it is closed for renovation at the moment and access is limited only to researchers.
The Museum of Turkish and Islamic Works and Arts (Turk ve Islam Sanatlari Muzesi) located in Ibrahim Pasa Palace at Sultanahmed Square, hosts another truly remarkable collection of over 15,000 manuscripts (from between 8th and 19th centuries). One of its 7 major sections is dedicated to manuscripts and calligraphy.
Another major Islamic manuscripts and calligraphy collection that’s on display is housed in Sakip Sabanci Museum’s Atli Kosk (Horse Mansion) in beautiful Emirgan neighborhood. The current exhibition focuses on Ottoman artifacts but also includes contemporary examples of Turkish art inspired by Ottoman calligraphy tradition. Majority of the manuscripts in exhibit are also available in digital format, allowing visitors to browse through all pages of the century old manuscripts on the ipads lent at the entrance.
Also, a limited number of manuscripts and samples of Islamic calligraphy are on display in various libraries, largest three of which are Suleymaniye Library (within the Suleymaniye Mosque complex), Topkapi Palace Library and Central Library of Istanbul University (within the Istanbul University’s Beyazit campus).
More information about Islamic manuscriptcollections in Turkey is available in English in the following site:
http://www.yazmalar.gov.tr/