I still haven't studied enough verbs to really be 100% sure. How do I say "My family is growing" in Latin?
Familia mea [some form of the verb crescere] est
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See this question about expressing the present continuous in Latin. While in English you use a form of "to be" with "growing", in Latin you do not use esse in any way. What you need is the present tense. From an English point of view, it covers both the usual present and the present continuous.
You got the subject "my family" correctly in the nominative case. What you need is just a form of crescere to go with that:
Familia mea crescit.