No. A bedroom is a multi-purpose room. The fact that people bathe and change in it does not turn it into a “bath-house,” and it is therefore obligated to have a mezuzah.[1] Likewise, even though the room contains a toilet, it also has many other uses and is therefore obligated.[2]
Moreover, some authorities hold that many of the halachos regarding toilets are not applicable to our modern flush mechanisms.[3]
Nevertheless, in order to recite Shema or berachot within four amot of the toilet, it should be covered fully with a cloth, even if it is totally clean and odorless.[4]
Since people sometimes undress in the room, the mezuzah should have an opaque cover, and if it is inside a bedroom of a married couple, it should have another covering aside from the opaque one.[5]
[1] Rema Y.D. 286:2.
[2] Sedei Chemed, Mem:119. Although Agur B’ohalecha (32:4:5) questions this ruling, he concludes that a mezuzah should be affixed without a berachah.
[3] Agur be-Ohalecha 32:4:5,134, citing Bei’ur Halachah 83:5. Cf. Chazon Ish O.C. 17:4.
[4] Mishnah Berurah 83:13.
[5] Magen Avraham 40:2