The purpose of minSdkVersion is to exclude platforms where you do not provide backward compatibility support.
In order to provide the proper libraries for your build you need to set a higher targetSdkVersion so the IDE or whatever knows what libraries to include when creating your APK.
It sounds like you don't want to target a higher SDK because some methods or objects may be deprecated or even unsupported. That's when you use support libraries, if necessary, for backward compatibility.