Dutton has come under fire from supporters for failing to win any of his first seven games in charge, and losing five of them.
In a video address to supporters Pomlett insisted that he believes in the work Dutton is doing and his talent as a coach.
He also respects his decision to stay with the club when he could have left in the aftermath of Darrell Clarke's departure.
Although Pomlett has backed Dutton, he has also insisted that results improve quickly – implying that his job could be under threat.
Pomlett said: "I'm not remotely happy with where we are in the league.
"A few weeks ago I genuinely thought this squad of players could have pushed to the play-offs. I actually think they should have pushed for the play-offs, but they didn't.
"I did not anticipate the loss of form and the drop into the lower half of the league. We should not be where we are.
"We are by no means the biggest budget in League Two and by no means the lowest budget in League Two, by a country mile.
"We're at least top half and I would argue probably top 10.
"We are a club on the pitch that is punching below it's weight, and I am unhappy about that.
"I still have enormous time and faith in Brian. I want to try and make sure he is successful, because he deserves to be.
"I believe Brian will have a big impact on this club and I have to be patient and hold firm with that decision for the time being.
"He knows that he has to win football matches because I will not allow us to get into a situation that I don't want to be in.
"Brian absolutely knows, as much as I like him and admire him, that I cannot sustain anything without any football wins in the coming couple of weeks.
"Brian will stay as the head coach of this football club, subject to him getting some wins very quickly."
To help support Brian in his new role, the club are now advertising for a director of football.
Interviews have already begun for the role, which will cover the academy and first team, as well as recruitment and mentoring.
Pomlett added: "I have made the decision to appoint an experienced football director, who will support the club going forward.
"It's a new role and one I've been thinking about for a year. I was going to do it for next season but have decided to make a move now.
"We will, in support of Brian, be appointing a director of football who will join the club as soon as possible.
"I'm not quite sure when the appointment will be made, but the interviewing process begins today.
"I will not make an appointment unless it's right."
Meanwhile, Pomlett has revealed that the club's directors have helped the club financially during the pandemic.
"We've had financial support from directors, me included," he said.
"I have directed cash into the club to help the club get through the pandemic. I'm not alone, other directors have done the same thing.
"All of the Covid-19 security was provided by one director for no charge.
"I have received support from the other directors and so has the club.
"Covid has hit us hard, no question, and we've managed through it successfully to this point, but I expect us to be impacted for a couple of years to come yet."
Regarding Pomlett's plans to return the club to League One and reunite them with the freehold of the Banks's Stadium, the chairman revealed that has been delayed by one or two years due to the pandemic.
He also reiterated his plan to take the club to the Championship and then sell to a new owner.