How to register my karate school for fights and tournaments?

Sounds like you are an independent school without a national organization. That makes it difficult to get connected. Find some of the tournaments in the area and talk with the promoters. They will be more than happy to give you information on upcoming events. Once you get connected, they will keep you on the mailing lists, which will probably be shared amount regional tournaments. They are always looking for people to give them money in the form of entry fees.

Make sure you understand the rules and regulations of the specific tournaments, as they vary from place to place.