What term is used to describe the process of converting bilirubin from its freely lipid-soluble form to the form that is tightly bound to the plasma protein albumin?

The term used is Conjugation. It is recycled of molecule hemoglobin that carries oxygen and a damaged or destroyed red blood cells are lysed. Separation of heme from globin yields biliverdin, which is converted to bilirubin and released into plasma and transported to the liver.