Age four is way too young to specialize in any sport. While my kids are not naturally athletically gifted, they have had some success in sports with practice (older two have had interest from DIII college coaches in one sport, eldest made it to honorable mention all-state in that sport).
We know one kid, still in HS, who is the most athletically talented kid we have ever seen. Around age 8-10, she could try any sport, practice for a few weeks, and be playing with the HS kids, holding her own. She played a variety of sports through middle school, but now has focused on one "lifetime" sport. She has verbally committed to a college that is great in both sports and academics.
Let the kid try a variety of sports. If you are aiming for college sports, you might try something other than gymnastics (not many schools have gymnastics teams). You also might find the kid is very talented at a young age, and while still very good in HS, she might not have Olympic level talent. I've seen that too.
Try to keep it fun and try other sports. Also, consider what sort of build the kid will have when grown. Does it fit gymnastics? My kids may have been more suited to the build for gymnastics, but chose sports where more height and weight would have been helpful.
