Only common fish - those that you can get from the common eggs - will remain healthy in the Fish Environment that you have at start of the game. If you cross-breed those fish, you will most likely produce fish that cannot survive indefinitely in a Level 1 environment. It sounds like your breeding has produced the most fragile hybrids, which will die at age 5 in a level one environment.

The only way to make sure that all of the hybrids that you produce will survive is to upgrade the Fish Environment in your tanks. You might want to read through some of the threads in the forum. There are a number of discussions and suggestions about how to get started successfully, as well as which fish are good to breed at the start of the game.