Immigrating legally means navigating a system both political parties say is broken. Here’s why it’s so difficult.

Jessica Mejía and her family had a good life in Colombia. They owned an electronics business and traveled to Houston every year, where they stayed with Mejía’s older sister Nataly, who had moved to Texas to study English at Houston Community College.

Mejía, her parents and her younger brother would stay for a month, eating out and shopping at Ross or Marshalls before flying back home to Medellín.

Everything changed in 2012, when a criminal group in Medellín approached Santiago, Mejía’s younger brother, demanding that he pay them for “protection.”...