Norris made her county debut in 2014, for Sussex against Warwickshire. She did not bat or bowl.[6] A year later, she was part of the Sussex squad that won the 2015 Women's Twenty20 Cup.[7] In the 2018 Women's County Championship, Norris had a strong season, taking 10 wickets at an average of 10.30 as her side was promoted from Division 2.[8] In 2019, Norris again had a successful county season, being Sussex's second-leading wicket-taker in both competitions, and hitting her maiden county half-century, scoring 59* opening the batting in a T20 victory over Warwickshire.[9][10][11] She was Sussex's joint-leading wicket-taker in the 2021 Women's Twenty20 Cup, with 7 wickets at an average of 11.71.[12] In the 2022 Women's Twenty20 Cup, Norris took eight wickets at an average of 7.87, including Twenty20 best bowling figures of 4/11, taken against Surrey.[13][14]
In 2017, Norris also played for Southern Vipers in the Women's Cricket Super League. She played one match, bowling three overs and dismissing Jenny Gunn for her maiden KSL wicket.[15] Ahead of the 2018 season, she moved to Loughborough Lightning, but did not play for them until 2019. Across her three appearances, she bowled three overs for no wicket.[16]
In 2020, Norris returned to Southern Vipers for the Rachael Heyhoe Flint Trophy. She appeared in all 7 matches, including her side's 38-run victory in the Final over Northern Diamonds. She took 12 wickets at an average of 17.91, including taking 4/45 in a match against Western Storm.[17][18] She was the joint-third leading wicket-taker across the whole tournament.[19] In 2021, she was the joint-second leading wicket-taker in the Charlotte Edwards Cup, with 13 wickets at an average of 13.30.[20] She achieved her Twenty20 best bowling in the tournament against Lightning, taking 4/14 from her 3.5 overs.[21] In the Rachael Heyhoe Flint Trophy, Norris took 11 wickets at an average of 30.09, including taking her List A best bowling figures, again 4/14, against North West Thunder.[22] In the final of the Rachael Heyhoe Flint Trophy, Norris hit her List A best of 40* as part of an unbroken stand of 78 for the eighth wicket with Emily Windsor to help her side recover from 109/7 to win the match, and the tournament, by 3 wickets with two balls to spare.[23] She also played six matches for Southern Brave in The Hundred, taking 3 wickets at an average of 22.66.[24] In 2022, Norris took two wickets in four appearances in the Charlotte Edwards Cup, and was Southern Vipers' leading wicket-taker in the Rachael Heyhoe Flint Trophy, with 12 wickets at an average of 19.75.[25][26] She was again in the Southern Brave squad in The Hundred, but did not play a match.[27] At the end of the 2022 season, it was announced that Norris had joined North West Thunder.[28] In 2023, she was Thunder's second-leading wicket-taker in both the Rachael Heyhoe Flint Trophy and the Charlotte Edwards Cup, with 10 and 9 wickets respectively.[29][30] She also scored her maiden List A half-century, with 51* against Southern Vipers.[31] She also played for London Spirit in The Hundred, taking two wickets in her six matches.[32] In 2024, she played 19 matches for North West Thunder, across the Rachael Heyhoe Flint Trophy and the Charlotte Edwards Cup, taking 13 wickets.[33][34]