One of the best and simplest ways to prove you're a competent programmer would be to just live stream yourself coding on twitch or YouTube. Both sites let you record and edit previous streams down to highlights you can organize into categories (e.g. database work vs frontend work).

Live streaming is also great practice for interviews as well, because you shout be narrating your train of thought and explaining pros and cons of your solution as you work on it, for both.

I did exactly this before acquiring my current job, and my boss thanked me for sharing the content and said that was the best possible interview process he could conceive of.