In empirical studies, multiple outcomes are often measured repeatedly over time, and interest frequently lies in studying the association between these longitudinal outcomes and a time-to-event outcome. Therefore, shared-parameter joint models for longitudinal and time-to-event outcomes have been developed. However, while such joint models in theory also allow for multiple longitudinal outcomes, they are often restricted to a limited number of outcomes due to computational complexity when fitting the models. To address this problem, we propose a new joint model, which is based on correlated instead of shared random effects, and for which a pairwise-modelling strategy can be used. In this approach, the longitudinal outcomes are modelled with (generalized) linear mixed models and the survival outcome with a Weibull proportional hazards frailty model. Instead of fitting the full joint model, this approach involves fitting all possible bivariate models, and inference is based on pseudo-likelihood theory. The main advantage of our approach is that there is no restriction on the number of longitudinally measured outcomes that are jointly modelled with the time-to-event outcome.