Is it permissible to bathe in the private bathroom of the house with some friends, knowing that it is designated for the family?

The basic principle is that it is permissible to use the bathroom in turns unless it leads to a religiously forbidden act. However, if it is a bathroom specifically for the wife and mother, then friends or others should not be allowed to use it except out of necessity, due to the potential for embarrassment or exposure of private parts. It is not permissible for your wife and your brothers to take turns using the bathroom if they are not her mahrams (unmarriageable kin). If possible, another bathroom should be designated for friends and guests to avoid embarrassment and blameworthy mixing. It is not permissible for you and your friends to bathe at the same time in one bathroom because it involves the exposure of private parts, based on the Prophet's (peace be upon him) saying: "Guard your awrah (private parts) except from your wife or what your right hand possesses." This applies to both ordinary and steam baths, as both require the covering of the awrah.