The Robins drew with Bournemouth at Ashton Gate on Boxing Day, meaning they have picked up just two points from their two games over the Christmas period.
They now face Port Vale and Leyton Orient, and Johnson wants wins in both games if his side are to overhaul Nottingham Forest and Scunthorpe at the top of the League One table.
"We now need to win our next two games," admitted Johnson.
"If we do that it will be eight points from the Christmas and New Year games, which would be satisfactory.
"We've conceded a bit of ground to our rivals and although the lads have shown a lot of spirit we have to start games better."