I went through the Turkey / Syria / Jordan loop a few weeks ago and you don't specifically need a Carnet, as the other fellows mentioned you instead pay an additional entry tax.
A friend traveling at the same time with no carnet paid a heap more in taxes though ( he was from the US ) than the cost of the Carnet, this may very where you cross but I'd suggest not, a tax is a tax etc.
The carnets I saw looked the same as mine, so the paperwork may have become standardised too.
So yes you can but it might be more expensive.
Cheers and enjoy, it was a great leg of the trip.