Kenan Karaman scored in stoppage time in Thursday's (AEST) friendly to cancel out Gian-Luca Waldschmidt's 81st-minute strike and leave the hosts winless in three games in 2020.
Julian Draxler and debutant Florian Neuhaus had also put Germany in front earlier in the match, only for Ozan Tufan and Efecan Karaca to hit back both times.
Germany also squandered one-goal leads in their 1-1 draw with Spain and Switzerland last month, and Low admitted it is an issue that his side must rectify.
"The problem has been around for a long time," he said at his post-match news conference. "This team, which has never played together like this, cannot be blamed.
"That has happened to the other players as well. I am disappointed and p***** off, that is the right word."
Germany made eight changes to their starting line-up from its most recent match and had limited time on the training pitch given the tight fixture schedule.
However, Low was not looking for excuses after a third draw on the spin in Cologne, where just over 300 fans were in attendance.
"Fatigue cannot be a reason, it is not an excuse or an explanation," he said.
"It's about losing control of the game, exploiting opportunities, simply losing the ball in zones where you don't always have to play across the pitch.
"When it comes to organising after losing the ball, we also have open positions that the opponent can use.
"When making the assignment, we have to say: In the box there is man against man, there is no space coverage. Concentration and mentality also play a role.
"[Karaman] was completely alone in the penalty area. Under no circumstances should that happen. We did this now and in a few other games. We have to talk about it.
"What can we do better when we have the lead? We then want to develop a mentality from this."
Draxler, given the captain's armband in the absence of Manuel Neuer, agreed with Low's assessment that Germany only had themselves to blame.
"We invited the Turks to score goals and again we didn't manage to get the result over the line," he told RTL.
"You just have to play it more confidently. It's disappointing. In the end, it's about winning the games - and we did not do that."
One positive for Low on an otherwise disappointing day was Neuhaus' debut goal, the Borussia Monchengladbach midfielder tucking in after combining with Kai Havertz.
"He had a good debut. He had a lot of good actions, especially when it came to goal, where he slipped very nicely into the corner," Low said.
Germany face Ukraine and Switzerland in their next two Nations League outings on Saturday and Tuesday respectively.