If a tattoo artist clearly messed up my tattoo, given their mistake shouldn't they be held responsible for removal costs?

Yes, you may be able to sue for the cost of removal (assuming they end up not paying voluntarily). You would have two possible grounds for your action:

1) Breach of contract, if they did not provide the tatoo you can show that you hired them to create;

2) Negligence, if they showed unreasonable carelessness in carrying out responsibilities (creating a tatoo) which they chose to undertake.

And if the skill level or ability of the tatoo artist was misrepresented, that could be fraud, providing a third possible ground for recovery.

If the cost to remove is several hundred dollars, you should probably sue in amall claims court, representing yourself. If the cost is several thousand, you should retain an attorney.