I think you're doing fine.
To me, the best way to convey this oddity of the academic publishing system is by sharing experiences and telling stories. If you get 20 academics in a room and tell the story "I once had 4 papers rejected in one week", I'd expect a combination of "Wow, that's great productivity to have 4 papers in submission at once!" and "Ha, that's nothing, one time I had 4 papers rejected in an hour!" It's just life in academia.
Now, I don't actually mean for you to assemble a room of 20 people to do this as some sort of intervention, but rather to reiterate this, share your own first-person and third-person stories, hopefully some that have a positive ending. And as more of a prophylactic approach, make sure there are social opportunities for your students to talk to each other and other researchers about these sorts of things. The group I work in has occasional "journal club" meetings where someone presents a paper that we all discuss, but especially in the good weather months we have these meetings outside, some people drink beer or eat ice cream, and people tend to hang out and linger afterwards. Or dinner parties. A lot of conversations about academia more broadly occur in those settings naturally.
Importantly, though, you also don't own or control anyone else's emotions. It can be difficult, especially for people with strong empathy, to be comfortable with others having negative emotions. You want to fix it! You want them to feel better! Sometimes, you can't. Having even 1 rejection at a time deserves a bit of a grieving process. As a mentor, you want to try to prevent that from becoming broader disillusionment and prevent it from turning into "giving up", but it's fine for it to hurt for a little while. Try to help set a schedule/goals for resubmission to keep a clock on it, but don't feel pressured to resolve this immediately, and don't put too much pressure on the next round needing to succeed. Also remember that you're not having an argument here, just present your own thoughts and feelings. If a student says they feel their work isn't good enough, if you disagree you don't need to tell them they're wrong. You can just say, "I'm proud of your work and proud to be your coauthor." It's fine if they roll their eyes, it'll sink in later.
Consider also some alternatives for spreading the work. Medicine is a bit slower on embracing preprints, but many journals are now okay with them: check the ones you plan to submit to, and if they allow then get your work up on medrxiv or similar, or biorxiv if the work is closer to basic science. Look for and suggest opportunities for presenting at conferences for your students: the networking opportunities are very important but they also give an opportunity to present work where you're face-to-face with the people evaluating it: that gives a better opportunity to defend the work and share the results compared to the faceless process of journal peer review. Look for opportunities for internal presentations to other students/faculty/clinicians as appropriate. These can happen before work is polished for submission, too: look for as many possibilities for feedback and sharing work as possible.