Forest began brightly and had a goal disallowed on eight minutes, with Taiwo Awoniyi flagged for offside after tapping the ball in.
Brentford almost took the lead on 52 minutes. Turner was closed down by Yoane Wissa, who intercepted the goalkeeper's clearance. With the ball heading towards goal, Willy Boly got back to clear off the line.
Moments later, Forest were reduced to 10 men, when defender Moussa Niakhate was shown a second yellow card for a foul on Wissa.
That was the 17th red card shown in the Premier League this season. At the same stage of 2022/23, after 68 matches, only five players had been sent off.
Forest pressed for an equaliser and were rewarded for their endeavour in the 65th minute.
The result moves Forest, who are yet to taste defeat in three matches at the City Ground this season, up to 11th on eight points.
Brentford, who have dropped eight points from winning positions this season, the joint-most of any team along with AFC Bournemouth, are a point further back and remain 13th.
Mauricio Pochettino’s side have failed to score in each of their last three matches and will be desperate for a much-needed win, while Fulham are unbeaten in their last two Premier League contests.