The blame starts and ends with the illegal Syrian regime.
Refugees are a result of the civil war that was started with the encouragement of the success in Tunisia, and the apparent success in Egypt. The US had to be dragged into this, because Obama could not see any reasonable group to support. It took almost two years before the US got seriously involved. See
https://en.m.wikipedia.org/wiki/Timber_Sycamore.
Public Opinion forced him into some token support with TOW missiles when the regime started to use tanks. CIA started training some rebels, but this was a total fiasco, producing maybe two dozen troops, which quickly deserted. Meanwhile the KSA and Gulf states provided massive support to the al-Nusra, to overcome the Iranian and Hezbollah mercenaries.
The US only got seriously involved in the fight against ISIS in 2014 and the Kurds were more receptive to US support.
The Syrian situation is more or less homemade in the Middle East.