Bridge has been with since 2001, joining from Publicis where she was in charge of the Woolwich account. She had previously held various marketing roles at United Biscuits.
The changes will see Marks & Spencer chief executive Sir Stuart Rose add the title of executive chairman to his business card, after the decision of present chairman Lord Burns to stand down from June 1.
Other departures include Guy Farrant, who steps down as director of retail after 30 years, and Simon Ratcliffe, director of logistics.
Nayna McIntosh will take over Bridge's responsibilities, Marks & Spencer said, with the title director of store marketing and design and a position on the executive board. She joined the company in 2001 from Asda, where she had been marketing director for the George clothing line.
Rose has committed to stay with M&S until July 2011, when he will retire. Other management changes will see Sir David Michels named as deputy chairman, starting June 1, and the promotion of Ian Dyson to group finance and operations director. Steve Sharp will continue as executive director of marketing, with responsibility for marketing, M&S Money and store design.
Lord Burns said that his main task as chairman had been to build the right board for M&S: "It is my view and the unanimous view of the independent directors, that placing Stuart in this new role creates the right leadership structure for the company.
"Stuart has the unique skills to continue the challenge of making M&S a world class retailer, and to develop the future leaders of the business."