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Relatives bathing together:
The Talmud prohibits one from bathing together with his father or step father, or father in-law, or sister’s husband [i.e. brother in-law]. Some Poskim extend this prohibition even against bathing with one’s brother, however other Poskim permit bathing with a brother.
Age of prohibition by child: It is permitted for a son to bathe with his father/brother until the child reaches the age or signs of adulthood. The age of adulthood is defined either by age [i.e. 11 years old for a girl], puberty [i.e. pubic hair], or maturity [i.e. the child reaches an age that he is ashamed to be unclothed in from of the parent].
The custom today: In today’s times, it is customary to permit bathing together, being that their private areas are covered in the bathhouse with a bathing suit and there is thus no worry of it leading to forbidden thoughts. [If however they are unclothed in the bathhouse, then from the letter of the law it is forbidden to bathe with the above relatives. Nonetheless, even today the widespread custom is to permit immersing in a Mikveh together with the above relatives, even though the private areas are not covered. This is accustomed even amongst G-d fearing Jews. Nevertheless, it is best to distance oneself from the above situation.]
From the letter of the law it is forbidden to bathe together with your father, father in-law, step-father, brother in-law and some add even with your brother. Practically, the widespread custom today is to be lenient in all the above.
May a father bathe with his young son?
It is questionable whether the above prohibition applies towards a young son. [Practically, the custom is to be lenient, as stated above.]
 Rama E.H. 23:6 and Y.D. 242:16; omitted from Tur and Michaber both in Yoreh Deah and Even Haezer
 Pesachim 51a
 Our Girsa of Rama ibid and Rama Y.D. 242:16; Gemara ibid; Mordechai; Gemara ibid, brought in Chelkas Mechokeik 23:5; Beis Shmuel 23:5
Other opinions: Some Poskim rule it is permitted to bathe with one’s step-father and that this is the correct Nussach in the Rama ibid. [Chelkas Mechokeik 23:5; Rif; Rosh Rashi; Ran; Aguda]
 Girsa of Rama Y.D. 242:16; Gemara ibid; Beis Shmuel 23:5 and Chelkas Mechokeik 23:5 Rif; Rosh Rashi; Ran; Aguda; Gr”a
Other Girsas of Rama: In our Girsa of Rama E.H. ibid he does not mention a father in-law.
 Rama ibid; Pesachim ibid
The reason: This is forbidden due to that it leads to forbidden thoughts. [Rama ibid] As one thinks of how he was born from his father, and how his wife was born from her father. [Rashi Pesachim ibid]
 Our Girsa of Rama E.H. ibid; Girsa of Beis Shmuel 23:5; Taharas Yisrael 23:24
 Chelkas Mechokeik 23:5 that this is the correct Nussach in the Rama ibid and so is the Girsa in Rama Y.D. ibid; Gr”a; Aruch Hashulchan 23:8; Girsa of: Rif; Rosh Rashi; Ran; Aguda; Gemara ibid.
 Beis Shmuel 23:5; Toldas Adam 6 in a story with Rebbe Zalman of Vilna and his father in-law, that as soon as he saw his father in-law in the bathhouse he left and did not return until his father in-law exited
The reason: As since the reason behind the prohibition is due to Hirhur, it is irrelevant as to whether he was there first. [ibid]
 This law is not mentioned explicitly in Poskim although is derived from the following other law mentioned: Prohibition to sleep together unclothed once the child reaches signs of adulthood. [Michaber Even Haezer 21:7; Michaber 73:3-4 and Admur 73:3 regarding Shema; See M”A 73:2 regarding seeming contradiction from Shema source; For possible answers of contradiction: See Perisha E.H. 21; Elya Raba 73:5, explained in previous footnotes] From the above source it can be understood that it is permitted for a child to see his parent of even opposite gender unclothed until he reaches the age/signs of adulthood. See however Aruch Hashulchan 23:8 that it is questionable whether the above prohibition applies towards a young son.
 See Poskim ibid in previous footnote.
 Rama ibid and Y.D. 242:16; Aguda Pesachim ibid
 Pischeiy Teshuvah 23:5; Toldas Adam 6 in a story with Rebbe Zalman of Vilna and his father in-law, that as soon as he saw his father in-law in the bathhouse he left and did not return until his father in-law exited, as the Heter of the Rama and Aguda no longer apply; Taharas Yisrael 23:24
 Pischeiy Teshuvah ibid; Aruch Hashulchan 23:8
The reason: Perhaps this is due to that the entire prohibition applies only against lounging around in the bath together with the relative and not towards simply being unclothed with the relative in the same building. [Aruch Hashulchan ibid; suggested answer in Mordechai, now although the Mordechai negates this approach, nonetheless perhaps this approach is accepted by the world.]
 Aruch Hashulchan ibid
 Aruch Hashulchan ibid
 Aruch Hashulchan ibid