Can I avoid putting my ex-husband on my tax return even if we're still married but he abused me and my kids?

[xmasbaby0]

If you are still  legally married he is not yet your "ex" husband.    But....if you lived apart for at least the last six months of 2020 and you have the children the you can file as Head of Household and do not enter anything about him on your tax return.   Does that apply to your situation?

Am I Head of Household?

https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household

https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household

If you qualify as Head of Household, when you enter your filing status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.

If you cannot file as HOH then your filing choices are married filing jointly or married filing separately.

If you were legally married at the end of 2020 your filing choices are married filing jointly or married filing separately.

Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will get the married filing jointly standard deduction of $24,800 (+$1300 for each spouse 65 or older)  You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit. 

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return. Some of the special rules for filing separately include: you cannot get earned income credit, education credits, adoption credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. Your limit for SALT (state and local taxes and sales tax) will be only $5000 per spouse. In many cases you will not be able to take the child and dependent care credit. The amount you can contribute to a retirement account will be affected. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. ( Community property states:  AZ, CA, ID, LA, NV, NM, TX, WA, WI)

 If  you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice.

https://ttlc.intuit.com/questions/1894449-married-filing-jointly-vs-married-filing-separately

https://ttlc.intuit.com/questions/1901162-married-filing-separately-in-community-property-states

https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separ...

[HelenC12]

Yes, if you qualify for the Head of Household (HOH) filing status. See Can a married person claim Head of Household filing status?

If you don't qualify for HOH and are using the Married filing Separately filing status, you won't be able to e-file. You'll have to mail in your income tax return if you don't included your husbands information.  Include a cover letter with your tax return and explain why the SSN, date of birth for the spouse are missing. 

See How do I print and mail my return in TurboTax Online?