In the field of medical imaging, the rapid rise of convolutional neural networks (CNNs) has presented significant opportunities for conserving healthcare resources. However, with the wide spread application of CNNs, several challenges have emerged, such as enormous data annotation costs, difficulties in ensuring user privacy and security, weak model interpretability, and the consumption of substantial computational resources. The fundamental challenge lies in optimizing and seamlessly integrating CNN technology to enhance the precision and efficiency of medical diagnosis. This study sought to provide a comprehensive bibliometric overview of current research on the application of CNNs in medical imaging. Initially, bibliometric methods were used to calculate the frequency statistics, and perform the cluster analysis and the co-citation analysis of countries, institutions, authors, keywords, and references. Subsequently, the latent Dirichlet allocation (LDA) method was employed for the topic modeling of the literature. Next, an in-depth analysis of the topics was conducted, and the topics in the medical field, technical aspects, and trends in topic evolution were summarized. Finally, by integrating the bibliometrics and LDA results, the developmental trajectory, milestones, and future directions in this field were outlined. A data set containing 6,310 articles in this field published from January 2013 to December 2023 was complied. With a total of 55,538 articles, the United States led in terms of the citation count, while in terms of the publication volume, China led with 2,385 articles. Harvard University emerged as the most influential institution, boasting an average of 69.92 citations per article. Within the realm of CNNs, residual neural network (ResNet) and U-Net stood out, receiving 1,602 and 1,419 citations, respectively, which highlights the significant attention these models have received. The impact of coronavirus disease 2019 (COVID-19) was unmistakable, as reflected by the publication of 597 articles, making it a focal point of research. Additionally, among various disease topics, with 290 articles, brain-related research was the most prevalent. Computed tomography (CT) imaging dominated the research landscape, representing 73% of the 30 different topics. Over the past 11 years, CNN-related research in medical imaging has grown exponentially. The findings of the present study provide insights into the field's status and research hotspots. In addition, this article meticulously chronicled the development of CNNs and highlighted key milestones, starting with LeNet in 1989, followed by a challenging 20-year exploration period, and culminating in the breakthrough moment with AlexNet in 2012. Finally, this article explored recent advancements in CNN technology, including semi-supervised learning, efficient learning, trustworthy artificial intelligence (AI), and federated learning methods, and also addressed challenges related to data annotation costs, diagnostic efficiency, model performance, and data privacy.