We investigate the connectivity properties of the random-cluster model mediated by bridge bonds that, if removed, lead to the generation of new connected components. We study numerically the density of bridges and the fragmentation kernel, i.e., the relative sizes of the generated fragments, and find that these quantities follow a scaling description. The corresponding scaling exponents are related to well known equilibrium critical exponents of the model. Using the Russo-Margulis formalism, we derive an exact relation between the expected density of bridges and the number of active edges. The same approach allows us to study the fluctuations in the numbers of bridges, thereby uncovering a new singularity in the random- cluster model as q <4 cos2 (π/√3) in two dimensions. For numerical simulations of the model directly in the language of individual bonds, known as Sweeny's algorithm, the prevalence of bridges and the scaling of the sizes of clusters connected by bridges and candidate-bridges play a pivotal role. We discuss several different implementations of the necessary connectivity algorithms and assess their relative performance.