Your question is confusing, you say you have been paying city tax that you don't owe, but you also say you are being billed for city tax you didn't pay.
In the state of Michigan, you owe city tax for the city you live in on all your world-wide income, if you live in a city with income tax. Turbotax can prepare some city returns but most can't be e-filed so you have to sign and mail the return. You also owe city income tax if you work in another city that charges tax, but only on income earned in that city. Your city of residence will give you a credit for tax paid to another city.
For example, you live in Lansing and you work in Detroit. You earn a salary in Detroit and you have a side gig in Lansing. You owe city tax to Detroit for the income you earn while working in Detroit and you owe Lansing tax on all your income. Suppose you owe $200 to Lansing but you pay $175 to Detroit. Lansing gives you a credit and you only pay $25 to Lansing so you don't pay double tax on your Detroit income.
Now, Turbotax will know where you live (from your zip code) but Turbotax won't know that you work some place else unless there is withholding on your W-2. Otherwise, you have to tell the program that you owe tax in another city. If you don't tell Turbotax that you work in Detroit, Turbotax won't calculate Detroit tax.
Although your question is not clear, I will assume you work in Detroit and live some place else and you have been paying city tax to the city you live in but not Detroit. In that case, once you calculate and pay your Detroit tax, you can recalculate your other city tax, apply the credit, and ask for a refund. You could do this in turbotax but it also looks like at least some cities have their own online site and use a common form. You might want professional help depending on how far back you need to go and how good your tax records are.
Here is a list of city tax returns that Turbotax can prepare. But most can't be e-filed.
https://ttlc.intuit.com/community/returns-and-forms/help/which-city-tax-returns-can-i-do-in-the-turb...
