He did baptize some along the way, but not many. Baptism was not an important part of Paul's ministry. He didn't remember baptizing more than a few people in Corinth. Peter, James, John and the other Jewish Apostles remembered baptizing thousands in Jerusalem. Because it was part of their gospel.
Most baptismal churches keep careful records of the baptisms of their members and families.
It's important to one's rights to participate in church matters and functions. Some baptismal churches believe it is necessary to eternal life. Paul, on the other hand, had no record of baptisms in any city where he preached his gospel. His memory brought back the names of a few people he had baptized, but that was all. No written records.