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Brailsford, 52, defended five-time Olympic champion Sir Bradley Wiggins after details of his therapeutic use exemptions (TUEs) were made public.
Wiggins was permitted to take a banned substance before major races and he has denied any wrongdoing.
UK Anti-Doping officials are looking at allegations of wrongdoing at Team Sky.
"I'm not proud of the way I dealt with this," Brailsford told the Telegraph Cycling Podcast.
"I have been through a lot of questioning and scrutiny. It's not comfortable. But if I didn't think we were doing it the right way I wouldn't be here."
Wiggins' use of the anti-inflammatory drug triamcinolone, which treats allergies and respiratory issues, was released by Russian computer hackers known as Fancy Bears.
TUEs allow the use of banned substances if athletes have a genuine medical need. Wiggins' TUEs were approved by British authorities and cycling's world governing body the UCI.
Former Team Sky cyclist Jonathan Tiernan-Locke has said that painkiller Tramadol was "freely offered" when he was riding for Great Britain at the 2012 World Championships.
British Cycling, the national governing body, put the allegation to the medic in question, doctor Richard Freeman, on the BBC's behalf, who denied it.
Brailsford said that Team Sky "try to be open and transparent as much as we can", but refused to comment on the contents of an alleged delivery that was said to have been made on the day Wiggins won the Criterium du Dauphine in La Toussuire in 2011.
"It's in the hands of [Ukad] to check that what is being said is the truth," he said. "I don't think it helps at this moment in time to say it was 'x'."
Team Sky boss Sir Dave Brailsford has said that his handling of the media following allegations against his team has made things a "damn sight worse".