Systems with slender fibers immersed in fluid flow appear in a multitude of applications such as stenting of blood vessels, cable subsystems to support off-shore floating structures, or terrestrial or submerged vegetation canopies. Yet, only recent approaches exploit the slenderness in the modeling process and represent the fibers by beam models as they arise from geometrically exact beam theory [1]. For the interaction of one-dimensional beam models (defined along space curves in three-dimensional space) with three-dimensional fluid flow, the mixed-dimensional character of fluid/beam coupling affects various aspects of the coupling algorithm. Besides the load and motion transfer between the beam and fluid domains as introduced in our prior work [2], the coupling iteration between the fields becomes even more challenging than in classical fluid/structure interaction. This is due to the slenderness of the beam, which is highly susceptible to the fluid flow. In this presentation, we will outline our mixed-dimensional coupling approach for fluid/beam interaction. We will particularly focus on the partitioned two-way coupling iteration between both fields. To this end, we employ a matrix-free Newton-Krylov (MFNK) approach and compare it to classical Dirichlet-Neumann coupling accelerated by Aitken relaxation. Despite the significant number of field evaluations for the MFNK approach, it delivers lower iteration numbers, but can also achieve a faster time-to-solution. We will not only assess the performance, but also demonstrate the robustness of our method though the simulation of large submerged vegetation canopies interacting with fluid flow. [1] C. A. Meier, A. Popp, and W. A. Wall. Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory. Archives of Computational Methods in Engineering, 26(1):163–243, 2019 [2] N. Hagmeyer, M. Mayr, I. Steinbrecher, and A. Popp. One-way coupled fluid-beam interaction: Capturing the effect of embedded slender bodies on global fluid flow and vice versa. Advanced Modeling and Simulation in Engineering Sciences, 9:9, 2022