This paper presents a finite element model for the analysis of crack-tip fields in a transversely isotropic strain-limiting elastic body. A nonlinear constitutive relationship between stress and linearized strain characterizes the material response. This algebraically nonlinear relationship is critical as it mitigates the physically inconsistent strain singularities that arise at crack tips. These strain-limiting relationships ensure that strains remain bounded near the crack tip, representing a significant advancement in the formulation of boundary value problems (BVPs) within the context of first-order approximate constitutive models. For a transversely isotropic elastic material containing a crack, the equilibrium equation, derived from the balance of linear momentum under a specified nonlinear constitutive relation, is shown to reduce to a second-order, vector-valued, quasilinear elliptic BVP. A robust numerical method is introduced, integrating Picard-type linearization with a continuous Galerkin-type finite element procedure for spatial discretization. Numerical results, obtained for tensile loading conditions and two distinct material fiber orientations, illustrate that the evolution of crack-tip strains occurs significantly slower than that of the normalized stresses. However, the strain-energy density is most pronounced near the crack tip, consistent with observations from linearized elasticity theory. It is demonstrated that the framework investigated herein can serve as a basis for formulating physically meaningful and mathematically well-defined BVPs, which are essential for exploring crack evolution, damage, nucleation, and failure in anisotropic strain-limiting elastic materials.