I will summarize the account of the Turkish historian Mehmed İpşirli that you can reach here. The link is in Turkish but plagiarism is to be avoided wherever we write, so there you go.
First, it is vital to draw a line between two kinds of executions: executions of nobles and executions of others. Ottomans followed an ancient Turkic tradition according to which the blood of the nobles was, in a sense, holy and therefore it was not to be spilled. Therefore, if a member of the dynasty was to be executed, this would be done by strangling, often using a bow. Hence the killing would take place without any bloodshed. If the one to be executed was not noble, then the usual method was beheading. It was also common for cellats (meaning the executor in Turkish and Ottoman Turkish), to hang the criminal near the crime scene (a burglar in front of the shop he robbed, for instance.)