It is difficult to say without seeing anything at all of the code you have, to understand what might be involved. Here is some speculation:
If you are using #[test] functions, they will run in an arbitrary order in multiple threads.
If you are mixing println!() and eprintln!() or dbg!() (more generally, stdout and stderr), depending on the environment, they might display out of order (e.g. the Rust Playground will display them entirely separately).
Please, show us at least part of your test program.
Buffering is a likely culprit. Are you using print! or println!? Standard out is buffered via a line break, which println! will include for you, whereas print! does not.