"I am bathing by the city."
Translation:A város mellett fürdöm.
This is NOT the definite conjugation. Please read up on the "-ik" verbs (I wish I knew where they are explained in the tips and notes but you can find a sufficient number of comments everywhere). Most "-ik" type verbs take the "-m" ending in the first person singular. But it still counts as indefinite. And intransitive.
There are some verbs that can take the definite form even without a valid reason, because some say they "sound better" this way. For me "fürdök" is still a better choice here.