Bridges are not only essential components of transportation infrastructure but also represent significant engineering achievements that impact the social and economic resilience of communities. Ensuring their structural integrity following major earthquakes is vital for maintaining post-disaster connectivity. As a result, the seismic design of bridges has evolved to incorporate performance-based approaches, with modern codes increasingly emphasizing nonlinear analysis methods grounded in displacement criteria. This study investigates the seismic performance of a three-span bridge using a displacement-based approach in accordance with the 2020 Turkish Seismic Code for Bridges (TKDY 2020). Special attention is given to the effects of soil–structure interaction at bridge piers and abutments. The bridge, measuring 92 m in total length and 17.50 m in deck width, features a superstructure of eleven 150 cm-deep prestressed precast I-girders topped with a 25 cm cast-in-place reinforced concrete slab. Both piers and abutments are supported on deep pile foundations. The bridge was first designed under the DD-2a seismic level using linear elastic analysis. A subsequent nonlinear time-history analysis was then conducted for the DD-1 seismic level to assess displacement-based performance. At least seven ground motion records were used in the analysis. Seismic demands for ductile elements were evaluated based on deformation capacities, while for non-ductile elements, strength demands were assessed using the mean of the maximum absolute responses from a minimum of 14 analyses. The results provide insights into the displacement demands and performance levels of key structural components, contributing to more resilient bridge designs under severe seismic loading.