Baptism should be a public event, but in Saudi Arabia a public baptism would be tantamount to a request for a beheading. Nevertheless, the event should be witnessed, though the number of witnesses may vary greatly. Baptisms done by John the Baptist were done outdoors, attended by great crowds of witnesses. The Ethiopian official whom Philip met on the road to Gaza seems to have been baptized near the deserted highway on which he was riding (see Acts 8:26–40). Saul of Tarsus (Paul) was probably baptized in or near the house where he was lodging, on Straight Street (Acts 9:10–19). Cornelius’s baptism was like Paul’s, in or near his own home (Acts 10:23–48).
These accounts are condensed. They do not tell us much about the nature of the ceremonies. They do note (1) that the person about to be baptized made a statement of faith, and, (2) that there was little or no delay between their confession of faith and subsequent baptism. Careful reading of the respective accounts reveals that witnesses were at every baptism. The baptizing was not done secretly.
Your account of your baptism raises other questions. Were there witnesses, if only as many relatives and family friends who could be crammed into the bathroom? How old were you—were you an infant or toddler? If your baptism followed your profession of faith and witnesses were present, I would conclude that being baptized again is unnecessary. If, however, you were too young to make a profession of faith or no one witnessed the baptism, I think the answer to your question is yes, you should be baptized again to publicly identify yourself as a follower of Jesus Christ.