Crosbie joins from CYBG – the parent company of Clydesdale Bank, Yorkshire Bank and Virgin Money – where she is currently group chief operating officer and executive director.
She is replacing Paul Pester who announced his resignation in September following the bank’s long-running IT failures.
Current executive chairman Richard Meddings will continue in the role until Crosbie’s arrival, at which point he will return to his previous position as non-executive chairman.
Crosbie joined CYBG in 1997 and has held a range of leadership roles within the business, including serving as acting CEO in the first half of 2015.
In May 2014 she was promoted to CYBG’s board and was appointed COO in January 2015.
Meanwhile, in September 2017 Crosbie was also appointed vice chair of the Confederation of British Industry (CBI) Scotland.
Commenting on the appointment, Meddings said Crosbie stood out among the candidates.
“With over two decades of experience, superb retail and SME banking expertise, and a genuinely open and engaging style of leadership, we have found an outstanding new CEO,” he said.
“What’s more, Debbie brings a clear challenger mindset to TSB and she shares our vision for what a strong challenger bank can achieve for customers.
“I’m confident that Debbie will have great chemistry with the TSB executive team and together they will drive our ambition to become the leading challenger bank in the industry.”
Crosbie added that she respected the bank’s values and culture.
“I’ve heard so much about the team at TSB and I can’t wait to get to know everyone,” she said.