Now, if that's not the case and your skin is just more sensitive than some people's, here are a few options. One, make your sessions shorter. If you can only handle 30 minutes worth, then do so. If you're working on a large tattoo, you may feel pressured to sit through longer sessions than your body can handle. You hold the money and you make the rules. If you need a break, tell the artist you need to stop for 5 minutes. If you need to go home, then go home and tell him you'll get more done in a couple of weeks.
If you're not sure you got the right person for the job then don't be afraid to search for a new artist to complete the tattoo. This should be a bearable experience at least and an enjoyable one at best.

