In this study, we proposed a new extension of the Haq distribution using an exponentiated-G family of distributions. The new distribution is named the exponentiated Haq distribution. The exponentiated Haq distribution improves the adaptability and flexibility of the underlying distribution and is thus applicable to a variety of datasets with such features as skewness, heavy tails, and high variability. Important properties, such as quantile function, mode, moments, incomplete moments, entropy measure, as well as order statistics of the proposed distribution have been obtained. The estimation of parameters of the exponentiated Haq distribution was conducted using six classical parameter estimation methods and an extensive simulation study was conducted to determine how the estimators performed. Real-world applications to radiation and COVID-19 datasets demonstrate that the proposed distribution outperforms several existing distributions, establishing it as a powerful tool for modeling complex data in diverse fields.