The lively Moses Odubajo raced onto a pass from debutant Jamar Loza to fire Orient in front in the 27th minute.
Then, after Tom Eaves squandered a golden chance to equalise for the struggling hosts, the Londoners added a late second with defender Scott Cuthbert credited with getting the final touch to a Dean Cox corner.
While Russell Slade was all smiles, these are worrying times for Shrewsbury manager Turner
His side have now lost five successive matches at Greenhous Meadow, failing to score in the last four.
"There's been a long period where this place has been a fortress for us," he said. "The one season we didn't lose a game here and the following season we didn't lose too many and always looked likely to get goals.
"But, unfortunately, these things happen from time to time. They go in cycles and we've hit a real bad patch at home where goals are obviously a concern.
"Sometimes, and it sounds simple but it isn't, it's just a case of finishing a chance and putting the ball in the back of the net."
Turner, whose side sit just two points clear of the relegation zone, added: "In patches we did okay and in patches we didn't work hard enough.
"The old saying is goals change games. We had a really good chance early in the game which given the home form recently and the state of confidence at home we needed to take.
"Second half we had a real go and for the first 20 minutes we looked brighter. We played at a better tempo and worked harder. We should have scored in that period.
"I'm disappointed in the manner we conceded the goals as that puts you on edge.
"I thought we were reasonable until we conceded the first goal and it was a bad one to concede. We shouldn't be conceding possession in our own half and letting them get right through at our back lads.
"You could see almost for the rest of the first half the confidence drained from the players.
"I know players can pass the ball better than we've shown there so that edginess came about. Everyone's thinking 'here we go again."