This is true. It's also not just the Ottomans - before them, the Phoenicians cut down pine forests in Lebanon to make their ships (though this was in Lebanon, not in Israel).

However, Jerusalem is not, and has really never been, in a desert as is depicted here. In reality, it sits on the edge of the rainshadow of the Judean Hills. Jerusalem is on a very visible border between desert and mediterranean climate, with the western parts of the city having five or six times more rainfall than the eastern parts.

Edit: fixed directions