In this work we introduce the notion of decisional width of a finite relational structure and the notion of regular-decisional width for regular classes of finite structures. We show that the validity problem for regular-decisional classes of structures is decidable. Building on this decidability result, we show that the problem of counting satisfying assignments for a first-order formula in a structure of constant width is fixed-parameter tractable with respect to the width parameter and can be solved in quadratic time with respect to the size of the representation.