It's not turing complete which is I think why it doesn't feel like a "real" programming language.

SQL is turing complete. A demonstration with recursive CTEs is done here[1].

[1]: https://wiki.postgresql.org/wiki/Cyclic_Tag_System

Oh wow. I stand corrected. Thanks!

Recursive CTEs exist; while I haven't investigated thoroughly, I would be surprised if they didn't make SQL turing complete.

More usefully, most databases allow you to write user-defined functions in other languages, including imperative SQL dialects.