you may share your type of light source and how do you switch the light on; e.g. by electricity of your light source ON or by a shutter or something else.
Just to see whether you light source can be the reason or the electric circuit of your detector.
You may also share the optical power impinging onto your detector as well as the kind of your detector.
(1) In the nature any process is not instant. There is some transition time needed to establish final state. At least it is still valid that maximal speed achievable is equal to the speed of light.
(2) In this case there several processes involved:
(a) Switching on process.
(b) light generation process (photo current saturation could be on of the reasons)
(c) photo current generation process (in silicon photo detectors time constant of few nanoseconds are common)
(d) Photo current read out process (Read out circuit capacitance related RC constant could be significant) .
(3) to find a good explanation for your curves you need to analyze all of them and identify the dominant one depending on circumstances.
As usual, to find good answers one need to make a lot of efforts. Some advice can help but majority of efforts is on you. Fortunately, light generation and detection nowadays is very important area in science and technology so there are a lot of literature where you can find some of answers.