The 1-0 defeat at home by Morecambe on Tuesday, in Webb's second match in charge, extended Orient's losing streak to six matches in League Two.
"With all the things the fans have seen and gone through it is hard to have optimism," Webb told BBC Radio London.
"There is light at the end of the tunnel. There are some good youngsters coming through here."
Webb named five players aged 20 or under in the starting XI which was beaten by the Shrimps - a defeat which means the O's remain 23rd in the table, one point from safety.
The east London side face fellow strugglers Cheltenham, Notts County and Newport County in their next six games.
"I understand the frustrations of not picking up any points in the last two games," Webb, 33, added.
"I have been proud of the players and the youngsters are looking like men out there. I want to assure the fans we are going to stick at it and keep going.
"They have a lot to be hopeful for with these players coming through, and the future of the club on the pitch is looking good."
Webb is Orient's fourth manager of the season and their 10th since the summer of 2014.
The club, who have been members of the Football League since 1905, have now lost 12 of their 16 league games at the Matchroom Stadium this season.
"I've have gone into this job with my own structure for the club in my head," Webb said. "Sometimes it takes a few defeats to get your message across.
"My first thought before the first day's training was that we need to get a bit of togetherness at this place.
"We need team spirit, camaraderie and fight to get the punters on our side a bit more."
Leyton Orient boss Danny Webb believes there are reasons for supporters to be positive despite their poor form.