Tribal Law and Order Reauthorization and Amendments Act of 2019 This bill revises public safety programs and services for Native American communities and modifies provisions related to Native American youth and juvenile justice. Among other elements, the bill requires the Department of the Interior to establish a five-year demonstration program for the purpose of conducting or adjudicating personnel background investigations for applicants for law enforcement positions in the Bureau of Indian Affairs (BIA); the BIA and the Indian Health Service (IHS) to enter a memorandum of agreement to make the IHS responsible for medical care and treatment of Native Americans detained or incarcerated in a BIA or tribal detention or correctional center, regardless of the individual's normal domicile; the Department of Justice to coordinate and provide oversight for its functions and programs to ensure a coordinated approach for public safety in Native American communities; the Administrative Office of the U.S. Courts to collaborate and consult with tribes to develop working relationships and to provide technical assistance and training for criminal defense techniques and strategies; Native American juveniles to be under the jurisdiction of the federal court system in delinquency proceedings in certain circumstances; and Interior to coordinate with specified agencies to assist tribal governments in addressing juvenile offenses and crime. The bill also reauthorizes various public safety programs and grants through FY2024, including those related to BIA law enforcement and judicial training programs, juvenile detention centers, tribal justice systems, control of illegal narcotics, and alcohol and substance abuse.