What you're describing is a pretty common problem and you can probably overcome it with the pick-up-put-down method. There are books on this method but the general premise is that you put her down "drowsy but awake" (meaning eyes closed or nearly closed). When she cries, even if it's right away, pick her up, calm her down, get her drowsy but awake again, and put her down. Lather, rinse, repeat. It takes stamina - this can go on 20 minutes to a couple hours - but it will help her learn to soothe herself, without CIO. She'll still know that you will come to her when she cries, which at 4 months, she needs.
Your daughter is still pretty young and may not be quite ready for this, but it's worth a try. I did this method with my son at about 6 months; he was the world's worst sleeper (up every hour every night, for months) and in about a week I got him down to two wakings a night; at 8 months he started sleeping completely through the night (7-5:30 anyway)!
I *highly* recommend the Baby Sleep Site. It's a web site dedicated to baby sleep and acknowledges that every baby is different and no single solution fits all. The woman who runs the site, Nicole, offers personalized, email-based consultation for a very affordable fee. That's what we did and it was WELL worth it. She'll analyze your situation and give you advice tailored to you so you don't have to read six books and figure out what to do from that.
PLEASE don't do CIO if you aren't comfortable with it, and certainly not at 4 months. I don't know that "crying is exercise a baby needs"; my son never had CIO and was always a strong, lean baby. Also be careful swaddling - it helps for very young babies but as they get older, they may actually be annoyed by the restriction. And once she's able to get her arms out or roll over at all, stop swaddling, because she may be able to get herself tangled or strangled with the blanket.