Domestic Violence Judicial Support Act of 2013 - Amends the Victims of Trafficking and Violence Protection Act of 2000, as amended by the Violence Against Women and Department of Justice Reauthorization Act of 2005, to authorize the Attorney General to make grants to state and local governments, Indian tribal governments, courts and other specified providers for: (1) supervised visitation and safe visitation exchange of children and youth by and between parents in situations involving domestic violence, dating violence, child sexual abuse, sexual assault, or stalking; and (2) court-related and child protective services workers education on the dynamics of domestic violence, dating violence, sexual assault (including child sexual abuse), and stalking.

Amends the Victims of Child Abuse Act of 1990 to: (1) ensure that by January 1, 2016, a court-appointed special advocate shall be available to every victim of child abuse or neglect in the United States that needs one; and (2) reauthorize through FY2018 the court-appointed special advocate program and the child abuse training programs for judicial personnel and practitioners.