Not normal.
What are you using as a chain cleaner?
What are you using as a lubricant?
Just based on what you wrote, I suggest you definitely increase your lubrication interval, and try reducing the cleaning interval.
Based on recommendations and info I've read about chain maintenance and the inner-workings of chain lubrication, I'm pretty much stopped "cleaning" (i.e. degreasing) my chains.
I stick to regular lubrication -- drop-by-drop, one roller at a time -- plus exterior wipe-downs when necessary.
In any case, lubricate more, and let us know what products you normally use.