Wales were much improved on recent showings and led 17-5 at halftime in a positive display, but errors at key moments allowed Scotland back in on Saturday.
Scotland moved to the top of the Six Nations table with 11 points from three matches, one more than France, who have a game in hand against Italy on Sunday.
The home side scored first-half tries through prop Rhys Carre and winger Josh Adams, but it was their improved defence that caught the eye, after they'd conceded 15 tries and 102 points in their first two games.
"It is gut-wrenching," Wales captain Dewi Lake said.
"Credit to Scotland, in their moments they were unstoppable. But the boys really fronted up and were dominant in that first half. Scotland had their moments in the second.
"We spoke a lot about staying in moments and switched on. For us, it is those key moments that are going away from us at the moment."
Scotland roared back in the second half and won thanks to tries from wings Kyle Steyn and Darcy Graham, flyhalf Finn Russell and Turner.
"We dug deep and did what we needed to do to win. Full credit to Wales, they came out with fight. We had to find a way to win ugly," Scotland's Australian-born captain Sione Tuipulotu said.
"I said to the boys at halftime we would need to find a little bit in themselves to come away with the victory. Several of the boys really out their hand up.
"It was not as sharp a performance as last week, but that was down to Wales' performance."
