The Reds have conceded 27 goals in their opening 14 games, leaving them sat at the bottom of the table on goal difference.
They lost 2-1 at Huddersfield Town last Saturday, inflicting a first defeat on Murray as interim chief following back-to-back draws.
But the former Mansfield Town manager feels that things are moving in the right direction, insisting that the goals they are letting in are more down to individual errors.
"In the last three games we haven't conceded as many chances," he said.
"But the chances that we are conceding are of high quality for the opposition.
"If you go through the goals in our last three games it is not a structural thing. It is coming down to individual errors.
"And the process of us getting better and moving to the next level is trying to limit the amount of errors we are making."
Murray has been in temporary charge at Oakwell since the shock departure of Daniel Stendel earlier this month.
And after two draws and one defeat in their last three games Murray feels things are moving in the right direction.
"In football you have two options when there are individual errors," he continued.
"You can take the individual out or you can work with the component and fix it to make sure the errors become rare.
"We are in that process and that isn't going to be fixed in the space of two to three weeks.
"From a structural point of view, we have been really pleased with how we have looked. We have looked a lot more solid, we look like we are going to score goals and we are creating chances.
"What we are finding at this level is, that in League One you can make a number of minor errors before the big error and you get away with it.
"But in the Championship you don't."
He added: "We are not going to go from not winning a game of football for a lot of games to overnight consistently winning football matches.
"What we have done is improve the performances and got everyone on the same page. We look like we can compete in every game now."
Barnsley host Bristol City tomorrow night, desperate to end their 13-game wait for a league win.
The Tykes beat Fulham on the opening day of the campaign but haven't won in the Championship since.
Bristol sit in sixth spot, in a congested top half were only six points separate first place and 12th.
Murray insists that every league fixture presents a challenge but is backing his side to get a result on home soil.
"It is a cliche but this is the Championship," he added.
"We have played teams at the top of the league and at the bottom and every team is a real challenge for us.
"On our day, especially at home, I would back us against anybody.
"And we will focus on ourselves and make sure we are right.
"They have got certain strengths but they have got weaknesses and it is up to us to exploit them."
Meanwhile, Murray has said he is "constant dialogue" with the Barnsley board about the permanent managerial position but is yet to have a formal interview.