Visa Overstay Enforcement Act of 2019 This bill imposes various penalties on aliens who overstay a visa or lawful immigration status. An alien who overstays shall be fined or imprisoned for up to six months, or both. Such an individual may not be admitted into the United States for 5 years, and may not be granted a visa for 10 years. For subsequent offenses, the alien shall be fined or imprisoned for up to two years, or both, and shall not be admitted into the United States or granted a visa. The Department of Homeland Security shall make case-by-case exceptions for aliens who overstay due to medical necessity, public safety, or national security reasons.