I've had a long battle with this with my younger son (now 7) because he compares himself to his older brother (10) and feels that he is not as smart because he can't read, do math, draw, etc. as well as his older brother, and so he doesn't even want to try new things if he can't be good at them right away. It has been so frustrating for me to watch him struggle with feeling that he is not as smart, because I was the younger child who thought that my older brother was brilliant and I wasn't as smart as he was.
We have worked very hard at pointing out how exceptional he is at some things, and his response has always been, "Yeah, but that's easy!". He just didn't get it that he was expecting too much from himself. This year I finally sat down with him and put on a whole big show about me being stupid and no good because I didn't know French, Spanish, German, Portuguese, Russian, etc. Of course, he rushed to console me and explained that it was just because I hadn't grown up in those countries, but he thought I could learn it if I tried. I laughed out loud and shouted, "Exactly!!", and then explained to him that anyone can learn to do anything if they try and put in the effort. It may be hard at first, but everyone has to learn things before they're good at them. This talk, along with the fact that this year he realized that he was better at something than his older brother, have really helped him to turn the corner and have a little more self confidence. Luckily my older son has been a great sport about making a big deal of his little brother's accomplishments as well, which has also been a great help.
I know this is for a little older age group, but it is the end of this issue that we have been dealing with for years. As far as the 4-year-old, there has been a lot of great advice here already, but one thing that I don't think has been mentioned, is see if some positive reinforcement from a peer of your daughter would help... Kids that age can be pretty easily prompted, and if one of her friends came over to do some art work or some project with her and then told her what a good job she was doing, it might sink in a little more with her than the same compliment from an adult. If you can enlist the help of the right 5-6 year old, who can take her by the hand and show her how to do some new things, it might encourage her to try it.
Good luck!