Conjugal visits, officially called extended family visits or family reunion programs depending on the state, are available in only four state prison systems in the United States: California, Connecticut, New York, and Washington. The federal system does not offer them at all. If your person is housed anywhere outside those four states, the program simply does not exist where they are.
Even in states where conjugal visits are permitted, eligibility is not automatic. Inmates typically need to be in good disciplinary standing, housed at an appropriate security level, and legally married or in some cases in a registered domestic partnership. The visits themselves are scheduled in advance and take place in designated private units on facility grounds, not during regular visitation.
Outside of the formal program, the reality of what happens in standard visiting rooms varies considerably by facility. Security staffing levels, supervision attentiveness, and the physical layout of a visiting room all affect how closely contact is monitored. In facilities where oversight is inconsistent, physical contact beyond what is officially permitted does occur. That is not an endorsement of it, just an honest acknowledgment of what anyone with prison experience has seen firsthand.
The formal conjugal visit program is the only sanctioned avenue, and it is limited to those four states. Everything else carries significant risk for the inmate including loss of visitation privileges, disciplinary action, and potentially new charges depending on the circumstances.