Just go with it. Really! I don't think a parent can teach independence. It comes naturally without having to push or cajole. But to encourage indepencence means you must invite dependence. Sounds strange, but only when your daughter feels safe and comfortable will she venture out on her own. She needs to grow the confidence within and your job is to support her. You are her compass point, her anchor.
My son (now 4.5 yo) rarely left my side at the same age as your daughter. I'd go to playgroups and he rarely left my lap. Kids would be dancing together and he'd sit and watch. I could tell that gatherings with lots of people were overwhelming for him. At around age 2 he'd simply ask to go home. So we went home. He didn't like being with anyone else but me and Daddy.
Fast forward to him as a 4.5 year old now and he's super-talkative and social. He's still a little cautious in large groups, but that's just his personality. He's quieter and introspective. He's not in preschool. He's with me and his brother all day and I did nothing to make him socialize more or push being with kids his own age. The only thing I did was respond to him needing me and I think it's that support that has helped him blossom socially.
But boy were those early years tough. I had a sling where I could carry my son on my hip and socialize with other folks at playgroups. We also did smaller playgroups in our home. I think my son was more relaxed in his own home. I'd have the same friends over so my son could get used to each person. That was better for him than the playgroups we went to where the groups of kids and moms was constantly changing.
Just ride it out and give her the support she needs. Not meeting her needs for closeness with you might cause her anxiety and have her cling to you more. Dependence is a good thing!
Good luck!