A lot of things floated through my mind when reading your question. (I upped you because you took the trouble to actually look at things and come up with such a question -- too rare here, I think.) One of them was charge storage in the BJT bulk and transit times. Another was to wonder how you'd set things up for measurement. Another was that you should check out your observations by also seeing if a spice simulator also would show the effect.
But as all these flew by, another much more obvious one arrived that probably explains the hogs share of what you are seeing and stopped me from wasting time on the rest. It's that a mere \$60\:\textrm{mV}\$ change in the base-emitter voltage would lead to a 10 fold change in the collector current.
Think about that for a moment. Say the LED current is \$20\:\textrm{mA}\$ at 100%. To go from 100% to 90%, the base voltage would change by perhaps \$2.5\:\textrm{mV}\$. With \$1\:\textrm{V}\$ per division, and noise, there's no way you'd see that. To go from 90% to 9% (nearly OFF), the voltage would drop by another \$60\:\textrm{mV}\$. Merely 6% of one division on your scope. And the LED is nearly OFF and therefore the \$V_{CE}\$ has risen to nearly it's maximum value now.
Of course the display would look like that. And I don't need to invoke charge storage and transit times, worry about your wiring or your hookups, or anything else. It's the expected behavior without such complications. Those complications may be there, too. But they aren't needed to explain your observation. I'm sure spice will confirm this.
You didn't set things up on the scope well enough to drill in deeper. The explanation above is enough to explain things at your current resolution. But, if you set things up to provide better precision (time and voltage) to narrow in further, it will turn out that my explanation above is no longer enough and you may be able to start pointing to further details which remain unexplained by this single principle alone.
Of course.
At that point, you are starting to dig into why BJT models are in fact more detailed and more complex and why there are so many added terms.
But at the level your display currently shows? That one basic principle is sufficient.