In the Gorakhpur publication, the description of the bridge construction is found in the 22nd Sarga (chapter) of the Yuddha Kanda. Furthermore, the 27th Sloka of the 25th Sarga mentioned in this question refers to the phrase “Agamya nagarIM laNkAm”, which describes Perumal and the Vanaras entering Lanka. Therefore, please verify the Sarga number you have provided.
Vibhishana suggested surrendering to the Ocean King. Accordingly, Lord Rama performed Sharanagati (surrender) to the Ocean King by observing a fast (lying on darbha grass). Seeing this, the Ocean King hesitated to appear before the Supreme Lord, thinking, He has performed Sharanagati to me; how can I go and stand before Him? and stayed away out of fear and reverence. Seeing that he did not come, Lord Rama asked Lakshmana to bring his bow and arrow. Just as he strung the arrow to shoot at the ocean, the Ocean King surrendered to the Lord. Furthermore, the Ocean King requested that the arrow Rama had strung be fired at his enemies, and Perumal did so.
Following this, the Ocean King suggested building a bridge over him, and Lord Rama had Nala construct the bridge. The bridge was built after permission was given to build it. No one was afraid after that point. Here, the one who was afraid was only the Ocean King. He feared two things: standing before the Lord and the possibility of the Rama-arrow striking him. Lord Rama was not afraid at any moment.
Note
Since there was a lack of clarity in the question you asked (regarding the citation), we have provided only the description of the Sethu Bandhanam from the Ramayana here.