Fiber lasers, the latest laser-cutting technology, are notable for their high process efficiency, cutting precision, and high cutting quality for thin materials. However, the quality of the cut significantly decreases when machining thicker materials. For now, this is a challenge for the metalworking industry. This study investigated the effects of laser power, cutting speed, and auxiliary gas pressure on the fiber-laser cutting quality of 4 and 6 mm thick S355JR steel plates. To evaluate the influence of cutting parameters on cutting quality, surface roughness, dimensional accuracy and cut taper were measured. A microscopic analysis of the laser cuts was performed, revealing the heat-affected zone, transition zone and unaffected base-material zone. Research results show that laser cutting is a complex process, and the correct choice of cutting parameters greatly influences the cutting performance and final quality. An artificial neural network was created and trained using the results from measuring the quality characteristics to achieve optimum cutting quality. The accuracy of the optimization model was assessed by control samples, which were cut using calculated optimum parameters. The actual values of the quality characteristics only slightly differ from the predicted values, showing that the optimization model is suitable for selecting cutting parameters.