How can I fix a toilet that is backing up with sink water?
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.