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Browse credit cards from a variety of issuers to see if there's a better card for you.
Credit cards ARE loans (not free money as many habitually lined up in this sort of debt believe) and many issuers often lump them together as such when considering applications. At lower scores (500-600 range) they will be looking to how you handle other types of loans. But yeah most working people are paying some kind of loan (car, student, house) in life so simply having those kinds of loans shouldn't be an issue. In the end 'types of credit' is only a small segment of the credit score they won't be rejecting you for 'lack of installment loans' when you have a few years good history with your cards.
After a while you start to get :
Limit requested exceeds banks lending policies... (meaning you have reached their internal cap of what they are willing to risk on you).