The reason the US cannot designate al-Qaeda in Syria as a terrorist group is because that would make the US legally responsible for having supplied a terrorist group with extremely sophisticated weapons and training. The US is arming and training terrorists in Syria, but instead of just getting out it is pretending that al-Qaeda is not a terrorist organization.
It might come as a surprise to many Americans that their government does not classify al-Qaeda in Syria as a terrorist organization. The reason it refuses to make the determination would shock them even more.
The Syrian franchise of the organization involved in the 9/11 attacks on New York and Washington has long gone by the name Jabhat al-Nusra (Nusra Front), and was sent into Syria by the head of al-Qaeda in Iraq, Abu Bakr al-Baghdadi. Baghdadi went on to launch the rival breakaway group ISIS, while Nusra maintained its status as al-Qaeda’s boots on the ground in the war against the Syrian government.