Use a drain auger on a toilet clog

If the sink is backing up into the toilet, then the clog is at or after the junction of the two in your drain lines. You'll have to snake the drain to remove the clog, and it will likely be beyond the reach of a standard toilet auger. You could remove the trap on your sink and attempt to snake it from there, or remove the toilet and run the snake down that drain.

If you do remove the toilet, you should wait for the water in the lines to recede so it doesn't flood your floor. If you go through the sink drain, you'll likely have a smaller drain line to maneuver the snake and it's possible the clog is further away, requiring a longer snake.

Slow flushing is not a symptom of a clog in the main line, that's not how toilets work.

You are probably thinking the pipe is full of water, so the toilet is having a hard time adding more, but if your pipe was full of water you would have water coming out of your bathtub! And every sink downstairs would be full of water.

A slow flushing toilet is simply an old, bad, toilet. It has nothing to do with the sewer line.

A sewer line is empty usually, unlike supply lines, sewer lines do not normally having anything in them, except for the few seconds after use. Even if the main line was slow (which it probably isn't), the small amount of water in the toilet would simply fill up the pipe above that part at a normal speed, then move down a bit slower - you would notice nothing.

Get yourself a new toilet, I like the Toto brand, you can buy one on Amazon for about $200.

If you had a clog the symptoms would be different: When you started using the sink it would work completely normally. Only after you ran the sink for a while (5 - 10 minutes even) it would start to slow down, and then it would stop draining and the water would just sit there. You would wait a long time and that water would very slowly drain down.