Not on turbotax.
While Turbotax can download your income information into a tax form (for participating employers only) that does not create a legal proof of income, since a person could just as easily create tax forms with fake income and print them out without mailing them.
You should always keep paper copies of your W-2s for at least 3 years, and preferably 7 years.
You may be able to get copies of your W-2s from your employer. Or you can get your Wage and Income transcripts from the IRS, this is a computerized record of what was reported to them. https://www.irs.gov/Individuals/Get-Transcript
Or you can get copies of your W-2s from the social security administration for a hefty fee (I believe $85 each).