Essentially the yarn isn’t knotting correctly (the loop) on the other side, the yarn seems to slide through the needle fine, it feels like the yarn is actually too thin, it is DK yarn and 100% acrylic, it also says its suitable for a 4mm needle.
Are you sure you threaded the needle correctly? Setting up can be a bit of a challenge. I did a punch needle workshop with a friend a while back and I remember getting started was an issue for most participants.
I attended a workshop on bunka embroidery, and I’m not sure how similar it is. We used a special thread that I believe was made from rayon. It was pretty elastic, almost curly, and “fluffed out” a lot so it was fairly easy to get it to stay after punching through.