In the US, I think "measure" is the term favored by "serious" (i.e. classical) musicians and professional music educators—it's the term they teach you in school. But "bar" seems to be the informal term that is favored by jazz musicians and others of their ilk.

There's the famous joke that comes from the Smothers Brothers:

Tom (sings): "Soap, soap, soap, soap, soap, soap, soap, soap."
Dick: "What are you doing?"
Tom: "Oh, singin' about eight bars…"