The phrase "bitter fight" means a conflict or argument that is filled with intense anger, resentment, and hostility between two or more people. Full definition
There can be bitter fights over who gets the car, the couch, and how much time and when each party can spend with the child or children. (divorcemediationonline.com)
As the Tremont Tribune reported in June, CIR / SEIU Healthcare had been engaged in bitter fight with St. Barnabas Hospital, whose administration has been blocking resident doctors» attempts to unionize. (bronxnewsnetwork.org)
As I write, Syrian Kurds in Kobane are engaged in bitter fighting against IS forces. (blog.politics.ox.ac.uk)