A novel family of statistical distributions, called enriched truncated exponentiated generalized family, is theoretically developed to model heavy-tailed data. One of the three-parameter sub-models of this family derived from log-logistic distribution is comprehensively studied. The statistical properties are explored, including moments and Fisher information matrix. In addition, tail-heaviness is studied using the tail-index approach. The method of maximum likelihood is used for parameter estimation, and existence and uniqueness of these estimators are shown. The flexibility of the new family is further validated by applying to the Norwegian fire insurance claim dataset. The goodness-of-fit measures are used to illustrate the adequacy of the proposed family of distributions. Furthermore, a backtesting procedure is conducted for well-known risk measures to assess the accuracy of the right tail fit.