No, this is not a normal reaction. I think you should ask about
copaxone. Avonex is so old! My neuro only perscribes it if you have been on it for a long time and don't want to switch. She just does not love it. Rebif is the same thing as avonex but a higher dose. So I would not do rebif if I were you as it seems like you would get the same reaction. It may be that you have a problem with interferons. That is why you might want to try copaxone. Don't let your doctor tell you that copaxone is not as effective because it is not an interferon. Some docs think this.
You are caught in a catch-22. As a person with MS, you want to avoid all infections as that causes an immune response that can exacerbate MS, but it seems that your treatment is causing those infections. I DO NOT have frequent infections and I am a teacher!!! I am exposed constantly!!!! I am almost never sick. Your goal is a balanced immune system that does it job!! You cannot suppress it so much that you are getting sick and then kicking your immune system in gear. An active immune response system creates T-cells an B-cells running around....you never know when those are going to cross the blood/brain barrier looking for some myelin to munch on.
I think if your MRI shows anything other than very very mild/slow progression then consider switching therapies. If you do switch therapies, you will have to wash out with avonex before you switch. This does leave you at risk for a flare. But you deserve to NOT be sick all the time! You also deserve a neuro that takes your disease seriously. Are you seeing an MS specialist? If you are not........find one!!!
Keep me posted kath. I want to know how you are doing.