The kitchen sink drains slowly or when the garbage disposal is on; the drain snake does not help

[icegeicege]

I hope someone can help me figure out what's wrong... last night, after dinner, I found the sink only drained when the garbage disposal was on, and then the water would back up when it was off and then drained slowly (it does drain!). This is the first I had problems with it, and I just got this new garbage disposal 6 months ago.

I tried the following:
- put the garbage disposal under running water to test whether water could go through it smoothly: yes.
- used baking soda + vinegar: this does not solve the problem;
- put ice in the sink and run disposal: this does not solve the problem;
- used a 25-ft electric drain snake to see whether anything blocks the pipe: three times, the head of the snake would carry black stuff and pieces of fish bones (I ate whole fish last night) with it.
- after I used the electric drain snake, I put three bottles of water into the trap to check whether the pipe is clogged, the water level was always at maybe 1/2 inch to the edge when I put water.

I'm running out of ideas now and don't know what's wrong with my sink so I come here for help! Thank you in advance! I attach a photo of my system under the sink to this post.

 

[Geofd]

Can you access the kitchen drain from below ?????? If it's still clogged you need a longer cable

 

[DanBurani]

Could be a grease restriction, did you have it jetted or just cabled??? My kitchen sink drained very slowly, i snaked it 25 ft. to the verticle, no difference. I cut the pipe open and found grease almost completely blocking the pipe. After jetting with cold water sink drains normally

 

[Geofd]

They won't jet only because the usually jet backwards from the main to your sink,and unless they had to they wouldn't jet usually a cable with a claw works well then alot of flushing with water