Ross Hannah's strike after a quarter of an hour had the Blues ahead at the break and the lead should really have been bigger after an encouraging 45 minutes.
Chester were slow out of the traps following the restart and the visitors levelled via Moses Emmanuel in the 53rd minute, and from that point on Burr's side struggled to get things going.
Johnny Hunt did have a good chance to put them back in front but it was the Ravens who went closest to finding a winner through Emmanuel and Adam Cunnington.
The game ended level, which dropped the Blues to 17th in the National League, and afterwards Burr said he felt the equaliser had affected his young side, which contained only one player over the age of 25.
"At the minute when we concede you can see the confidence draining from the players and I said it to them again they have got to deal with that," said Burr, who reckoned his side had a decent penalty shout turned down for handball in the first-half and another when Hannah was bundled over in the second.
"When you're not winning you get criticised and you've to deal with that but it's something some of these young lads are not used to and they clam up a little bit. I'm not saying they don't want the ball but they are frightened of making a mistake when they are on the ball.
"When Bromley got the equaliser, after that you could see that bit of anxiety coming back into us and that's what happens.
In pictures: Action from Chester's 1-1 draw with Bromley
"We looked through the stats of crosses into the box in the first-half and we should have added to the one that we scored.
"I thought we knocked it about quite well really particularly in the first-half but when you're on top you've got to take those chances.
"I felt Ryan Lloyd should have got in there on a couple of occasions, James Alabi and it's stuff we've been working on during the week with Danny O'Brien getting his crosses in and John Hunt was getting crosses in as well for us.
"We felt as well decisions are going against us a bit with the handball and even in the second-half we thought Ross had been brought down for a penalty but we're not getting that rub of the green at the minute and that's what happens when you're struggling."
Chester seemed on course for a much-needed three points when Hannah opened the scoring on 16 minutes, running onto an O'Brien pass and showing a cool head to fire past Alan Julian for his 20th goal of the campaign.
Hunt had the clearest chance of the second period but sliced his shot wide and Burr admitted the draw was a missed opportunity as the Blues strive to keep themselves clear of the bottom four.
He said: "I though the goal was a really good move. The best move of the game probably was the one where we got John Hunt in during the second-half and just as he was about to put it into the net you could see it bobble away from him so we're disappointed.
"We don't want to lose, it's a point and that's 40 points, It's a game that everybody here expects to win and we expect to win but Bromley have come here on the back of some good results.
"In the end it's a point but it's two dropped for me."