SciShow explains a couple assumptions about splinters you might have wrong. In showing how splinters don't necessarily get ejected from your skin automatically, they also show wood splinters can be much worse than a glass or metal splinter.
The belief might stem from the idea that wood being pliable and natural might make a wood splinter less intrusive and problematic compared to glass, which is all terrifying edges. However, that's not the case. Organic materials like wood, flower thorns or cactus spines cause more inflammation than inorganic materials. Additionally, organic materials are more likely to be carrying fungus or bacteria that could result in problems like tetanus or rose gardener's disease.
For an in-depth look at how this happens and how your body responds to a splinter that's left in your skin, watch SciShow's explanation above.