Advanced structures, which require high stiffness-to-weight ratios, often rely on curved shells to efficiently distribute loadings by exploiting their intrinsic membrane-bending coupling. Typically, their geometrical design is carried out using NURBS splines that, in the simplest case, map a square domain into a curved surface. For more complex geometries, the immersed boundary approach involves using lower-dimensional splines to define trimmed external or internal boundaries. In Isogeometric Analysis (IGA), the same splines used in the definition of the map are employed to construct the approximation space, resulting in trimmed shell elements at the boundaries of the structure. While conceptually simple, this approach has drawbacks. First, Dirichlet boundary conditions can only be imposed in a weak sense. Additionally, the structure's boundary is where stress concentration and other localized phenomena are likely to occur. Therefore, to accurately capture these behaviors without excessively increasing computational costs, local refinement strategies are required. This is challenging with classical tensor-product-based splines. In this work, within the context of isogeometric analysis, an Immersed Boundary-Conformal Method is employed to conduct the linear static analysis of laminated shell structures. Following this approach, a local IGA layer is created in the proximity of each edge, either conformal to the original boundary of the surface or resulting from a trimming operation. The main advantages of the proposed method are that these boundary layers allow for straightforward local refinement and a strong imposition of Dirichlet boundary conditions. The main patch and the boundary layers are linked together with an Interior Penalty method that ensures optimal convergence rates. As usual in immersed boundary approaches, tailor-made integration rules are required. Here, accurate integration relies on a high-order reparameterization of the trimmed elements. To demonstrate the capabilities of the proposed approach, several results are provided. A comparison between the solutions obtained with benchmark references validates the method.