You think The Turk tried to kill Don C. without at least the tacit support, and maybe more, of Barzini and Tattaglia?

I agree with BOB in his post above. I don't think there is any way That Sollozzo made this decision on his own.

How could he know for certain that the Barzini-Tattaglia alliance was willing to risk an all-out war with the Corleones over the drug business?

He would have had to check with Barzini first. If Barzini told him, "Well, too bad Virgil old pal. Looks like we're not going into the drug business", there's no way Sollozzo tries the hit. He'd be writing his own death warrant.

Remember Sonny's words to Tom (from the book, I think), something to the effect that to avoid a war, the other families have to "give me Sollozzo".

How did Sollozzo know they wouldn't?

I don't think this was a case of Barzini knowing that The Turk "would try to kill Vito."

More like Barzini knowing that Sollozzo would want to. Perhaps I'm splitting hairs here, but I think there is a difference.

And as Vito said, Barzini was in on it. All along.