He fills the role vacated a year ago by Myfanwy Moore, reportedly with immediate effect.
One of the most senior roles in the comedy-focused company, Simpson will report directly to Peter Salmon and Lucinda Hicks, executive chairman and CEO, respectively, of Tiger's proprietor, Banijay UK.
Salmon said: "David is an award-winning talent who has cut his teeth working with some of the best new people in the business. We know the world needs to laugh right now and there is a huge opportunity to build on Tiger's reputation for world class comedy and to find the next big names from all corners of the UK."
Banijay was founded in Paris in 2008 and has become a multi-national conglomerate of formerly independent production companies. 2020 saw Tiger's previous parent, Endemol Shine Group, acquired by the outfit.