Is it proper not to permit a child to wear tzitzis until he is potty trained? I heard of such a custom, and I was wondering what the reason is for this. People do bring a talis katan into the restroom so why should a child be different?
The obligation to train a child in wearing children begins from the age at which he is able to dress himself in the talis. According to Rema, the age is only from when he knows how to wear two sets of strings in front of him, and two behind him, and to hold the tzitzis during krias shema. Different ages are given for this, but the principle halachah depends on each child.
Yet, some write that a child should wear tzitzis from the time that he can speak, or from when he is three years old. No distinction is made concerning the potty training of the child. Yet it is possible that this custom comes from the wish to give due respect to the tzitzis.
Although, as you note, we do brings a talis katan into the restroom, we only do this because it would be a great burden to have to take it off the whole time. Where possible, it remains preferable to give proper respect to the tzitzis. However, Shaarei Teshuva implies that there is no problem in giving a young baby tzitzis.
Sources: See Gemara, Succah 42a; Shulchan Aruch and Rema, Orach Chaim 17:3; Shaarei Teshuvah 17:2. Concerning entering the restroom, see Shulchan Aruch 21:3; Taz; Kaf Hachaim 21:13; Yabia Omer vol. 2, no. 1, sec. 7