The result should have been more emphatic for Barcelona, who wasted numerous chances.
Turan impressed in attack, but perhaps the abiding memory of the Turkey midfielder's display was his first-half miss from eight yards.
Messi shot inches wide after typically drifting past two defenders, and also fired over the bar. Suarez, his partner in attack, thundered a volley over from eight yards and also had an effort from two yards cleared off the line by Eneko Boveda in stoppage time.
Bilbao will rue missing the few good chances that came their way. Midfielder Raul Garcia saw his strike narrowly drift beyond Ter Stegen's post and Inaki Williams' poked effort was saved by the Barca keeper.
Ter Stegen was also fortunate not to be left red-faced in the first half when his pass out of defence was fired straight back at him by Spanish midfielder Benat.
Las Palmas score nine goals in two games to go top
Earlier on Sunday, the Gran Canaria club beat Granada 5-1 as they leapfrogged Real Madrid to go top of La Liga.
Summer signing Kevin-Prince Boateng scored for the second successive game, having netted on his debut in the 4-2 win over Valencia last week. Nabil El Zhar scored twice in the emphatic win.
In the other early match, Sporting Gijon drew 0-0 at Alaves and the late game between Villarreal and Sevilla also ended goalless.