In statistical modeling, bivariate models are essential, especially when examining data with two associated variables. Bivariate distributions capture dependencies between variables, offering a more realistic depiction of real-world phenomena compared to univariate models that treat variables independently. This is particularly important in domains where variables frequently show non-trivial correlations, such as environmental science, reliability engineering, medicine, and finance. This motivates the proposal of a bivariate distribution that uses the generalized Farlie-Gumbel-Morgenstern (FGM) copula and Weibull marginal distribution, referred to as the GFGM-WD. The GFGM-WD describes bivariate lifetime data with weak to moderate correlation between variables. The suggested model was employed to investigate the reliability of dependent stress-strength models. Several properties of the GFGM-WD were derived, including the product moment, the coefficient of correlation between the inner variables, and the conditional expectation. Additionally, the statistical characteristics of the concomitants’ k-record values from the GFGM-WD were discussed. We ran comprehensive Monte Carlo simulations to assess the suggested distribution’s performance and used the maximum likelihood estimation and Bayesian methods to estimate its parameters. Finally, the distribution was tested on two actual medical datasets, showing that it outperformed other pre-existing bivariate models in terms of fitting accuracy.