Timely Departure Act This bill requires certain nonimmigrants to secure a visa with a bond. Under the bill, an applicant for a covered visa category must provide a bond (or cash payment) of at least $5,000 and no more than $15,000 according to rules to be established by the Department of Homeland Security (DHS). Covered visa categories include those for visitors for business or pleasure and exclude those for foreign government officials and crime victims. The bill also specifies that a nonimmigrant seeking asylum or withholding of removal while present in the United States must apply for asylum or withholding of removal by the expiration date on their visa to remain eligible for asylum or withholding of removal. The bill provides that a covered nonimmigrant who overstays the term of their visa forfeits the full bond, is subject to removal, and is ineligible for any lawful immigration status (including adjustment of status) for at least 4 years and up to 12 years. Forfeited amounts may be used solely for detention facilities for non-U.S. nationals ( aliens under federal law) and international transportation for non-U.S. nationals ordered removed from the United States. The bill specifies that DHS, in implementing this bill, may only issue rules regarding (1) the collection and retention of bonds and cash payments, (2) the notification of the Attorney General regarding a nonimmigrant's failure to timely depart, and (3) preventing the circumvention of the bond requirement.