This paper presents a novel adaptive crack-tip extended isogeometric analysis (adaptive CT XIGA) framework based on locally refined B-splines (LR B-splines) for efficient and accurate fracture modeling in two-dimensional solids. The XIGA method facilitates crack modeling without requiring the specific locations of crack faces and enables crack propagation simulation without remeshing by employing localized enrichment functions. LR B-splines, as an advanced extension of B-splines and NURBS, offer high-order continuity, precise geometric representation, and local refinement capabilities, thereby enhancing computational accuracy and efficiency. Various local mesh refinement strategies, designed based on crack and crack-tip locations, are investigated. Among these strategies, the crack-tip topological refinement strategy is adopted for local refinement in the adaptive CT XIGA framework. Stress intensity factors (SIFs) are evaluated using the contour interaction integral technique, while the maximum circumferential stress criterion is adopted to predict the crack growth direction. Numerical examples demonstrate the accuracy, efficiency, and robustness of adaptive CT XIGA. The results confirm that the proposed framework achieves superior error convergence rates and significantly reduces computational costs compared to a-posteriori-error-based adaptive XIGA methods, particularly in crack propagation simulations. These advantages establish adaptive CT XIGA as a powerful and efficient tool for addressing complex fracture problems in solid mechanics.