Jason Tovey kicked 14 points for the Welsh region but with Will Harries' try their only score they couldn't stop the Warriors - who ran in three tries through Bernardo Stortoni, DTH van der Merwe and Colin Shaw - taking the points.
Dan Parks' 12 points helped seal the Glasgow outfit's win and leaves them third in the table while their Welsh opponents sit sixth.
And Turner was less than impressed with his side's poor showing.
"It wasn't good enough, we made a poor start after we talked about not doing that here," said a disappointed Turner. "We were beaten in the contact area and everything we did to the Ospreys they did to us.
"But I was pleased with young guys like Toby Faletau as well as Andrew Coombs, but you've got to question some of the senior players.
"We were building towards a bonus point, but we coughed the ball up and then conceded a try at the other end.
"We know what we've got to do now and that's beat Edinburgh at home in our next game."