We prove an analogue of the Kauffman-Murasugi-Thistlethwaite theorem for alternating links in surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. The proof holds more generally for links admitting adequate diagrams and the key ingredient is a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. This result extends the first and second Tait conjectures to alternating links in thickened surfaces and also to alternating virtual links.