I live in Ohio and work in one city and live in another can I just have my employer just take out the taxes for the city I live and not take care of the city I work in.

If you live in one Ohio city but work in another, you get credit from the city you live in for the tax withheld from the city in which you work.  If there is a difference in the tax rate between those two cities, then you do pay the difference when you file your tax return.  Ohio does have a complicated system in which there are city taxes, RITA, and school district taxes.  Be careful when you prepare your OH tax return. Enter the information from your  W-2 very carefully.  Filing for those city, RITA, or school district taxes will not be automatic.

https://ttlc.intuit.com/questions/1899325-which-city-tax-returns-can-i-do-in-the-turbotax-state-program

https://ttlc.intuit.com/questions/3976472-did-turbotax-file-my-rita-for-ohio

https://ttlc.intuit.com/questions/3537370-how-do-i-pay-my-ohio-school-district-tax

https://ttlc.intuit.com/questions/4405762-when-i-e-file-ohio-taxes-do-my-local-rita-taxes-also-get-e...

If you live/work in a RITA location, you will probably find that is easier to file on the RITA site instead of trying to use TurboTax for RITA.

RITA Web site:  http://www.ritaohio.com/

No, there is nothing wrong. Your employer is required to withhold for the work location city.  Employers are only required to withhold for the work city,  although many will voluntarily withhold for the resident city, in addition (not instead of)

Any Ohio city, with an income tax, is allowed to have withholding taken out of your pay while you worked in that city. Your employer is required to do that withholding. You can't get a refund unless it was improperly withheld, i.e. you didn't actually work there for all or part of the time that your employer was withholding.  If you live in a city with an income tax, you are subject to that city's tax as well. The city you live in will usually allow a credit or partial credit for the withheld tax you paid to the work location city. Employees of most big companies, in Ohio, seldom owe any city tax, at tax time (but they still have to file a residence city return), because the employer holds out exactly the right amount for both work and resident city, taking into account the credit. If your employer is unwilling to do that because of the paper work hassle, you just have to deal with it. It's a bigger hassle for small employers than it is for the employee to just claim the credit on his resident city return.

Note: 

School District (SD) tax is completely separate from city tax. You do not get a credit for city tax withheld on an SD return and vice versa.

Only school district residents file a return &  pay the tax. Non residents who work within a taxing SD are not subject to withholding. If SD withholding was mistakenly withheld, your residence SD will not give you a credit. You have to file a 2nd SD return to get a refund.

You have to pay both for the city you work in and the city you live in. They are correct with taking out the taxes you work for and you will probably have to pay the city you live in at tax time