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Colin Charvis, Joe Bearman and Rhys Thomas touched down for the hosts, with two conversions and a drop goal from Ceri Sweeney while Nicolas De Gregori and Dion Kingi crossed for the visitors who had Marius Goosen to thank for the remainder of their points.
Turner said: "They were a physically big side and tended to play in the right areas. We could not break them down and, although we got a lot of things right, we got as many things wrong and that was the story of our afternoon."
Treviso skipper Goosen's fourth penalty sealed the game for the Italians deep into injury-time, and Turner added: "They had a good line-out and you could probably see near the end that, as we did not kick it out far enough, they got their driving going from the line-out and got the penalty.
"I'd like to look at the penalty count but usually if the set-pieces are good, Benetton as they were from Treviso, you tend to give away those indisciplines.
"But as many things that were good from us, there were as many bad and you can't win European games like that.
"I don't think we ever got control of the game even when we dropped the goal to possibly win it - I think there was four minutes left on the clock then - which was the same as last week.
"In those last magical four minutes, we don't seem to be composed enough to go on and win those games. So it is disappointing but we've played a good European side today. People may think we have come up short and we are extremely disappointed not only for ourselves but also four our supporters."
Former Springbok Smith, meanwhile, who was a full-back with Newport eight years ago, was watched at Rodney Parade by the former benefactor, office furniture multi-millionaire Tony Brown, which, he said, thrilled him.
But the win was vital for the Italians and Smith said: "I must say I'm very happy. In the end, it could have gone either way as both sides played some good rugby.
"We are growing as a team and we are a new group of players. We have new ideas and, in two years' time, we want to compete in the Heineken Cup so we are going to use this experience to go forward.
"Our aim at the moment is to earn the respect of others in Europe and what helps us is that every game we play is a growing phase. We have our own objectives and winning is not always the objective of the game."