Employer reported my bonus on form 1099-Misc box 3 [other income]. Is this correct? Can I avoid self employment tax?

Your employer is incorrect.  All monies paid to an employee must be included on the W-2 and are subject to both the employee share and employer share of social security and medicare tax (7.65% each).

Ask the employer for a corrected W-2. If they won't do this, in Turbotax when entering the 1099 there is a page of special circumstances and one of the boxes you can check is "this is from my employer and should have been on my W-2".  Turbotax will prepare the required extra form to charge you 7.65% employment tax instead if 15% SE tax. 

By using the 1099, and possibly by changing the name, the employer is trying to avoid certain costs (social security and medicare tax, unemployment taxes and workers comp insurance, and other costs that might be based on total wages.)  Possibly on purpose?  Possibly simple ignorance.  If on purpose, be on the lookout for other shady practices you might want to disassociate yourself from.

Since it's from your employer, it really should be on your W2 with the correct withholding taken out for social security and medicare tax.  I would start by requesting a corrected W2.  If you get no where with that request, I would enter it as Other Reportable Income under the Miscellaneous Income section.

Follow these steps: 

1. Sign in to TurboTax Online, click Continue your return, then pick up where I left off (if you are using TurboTax CD/ Download, then just open your return on your computer);
2. Click on Federal Taxes at the top of the screen;
3. Go to Wages and Income;
4. Toward the bottom, you will see a box to click that says Add More Income.  
5. Scroll down the page to the last section titled: Less Common Income and click show more;
6. Select start or update next to the last topic titled Miscellaneous Income, 1099-A, 1099-C;
7. Scroll down to the last option titled Other reportable income and select Start or Edit;
8. Click Start next to Other income not already reported on a Form W2 or Form 1099;
9. Click YES to Did you receive any other wages;
10. Continue past Wages earned as a Household Employee, Sick or Disability Pay, and select YES again to the question Did you earn any other wages?
11. Select Employee Compensation that was not reported on a W2;
12. Continue answering all questions.  The Reason Code you should use is H;
13. Then click Done.

Thank you for your response. I did feel like something wasn't right, especially since this was a "corrected" form. I will contact my employer to ask for a  corrected W-2 and not 1099.