For us, familiarity and gentle persistence was the key.
My son was 4-1/2 when we adopted him from China. He had never slept alone in his life. He couldn't tolerate being in a room alone. He was terrified of animals. We have a video of him at about five; a tail wagging golden retriever wanders close to him as he and his dad are playing on the front lawn, and when he sees the dog he practically levitates up his dad's leg into his arms.
At first we let him sleep in the same bed with his 6 year old sister. Then we moved him to a bunk bed above hers. He didn't want to go, and we let him do "sleepovers" with her on weekends, but we insisted that he needed to be in his own bed. We didn't scold him for being afraid, but we told him he needed to be a big boy. Then we had the house remodeled and added a bedroom for each child. He didn't want to move to his own room but we tried to show him the benefits. Again, we let him do sleepovers in the bottom bunk of her bed on weekends. Now, at age 12, he's finally decided he likes having his own space, though he still won't close the door.
He's gotten to the point where he will (reluctantly) tolerate the door being mostly closed when he uses the bathroom or takes a shower, but he never "remembers" to close the door on his own.
At first he was terrified of all animals (he literally ran away from a kitten at a birthday party, screaming). We got a German Shepherd puppy and by the time she had grown into a 90 lb dog he was fearlessly rolling about and playing with her. We rescued a kitten from a shelter and now he plays with the cat. He doesn't like animals whose species he hasn't been exposed to, but familiarity seems to be the key to reducing his fears.
We tried to strike a balance between supporting him and pushing him toward independence. Try giving little pushes and being very calm about the fact that she is afraid. Everyone has things they are afraid of but we all try to move toward independence from those fears. For some it takes longer. Don't rush her, but don't let her stall. Give her night lights and soft music as companions in her room instead of your presence. As she rides her bike try making little excuses to leave for a minute, then make the time stretch out longer and longer. Once she learns that nothing bad is going to happen when you are not there, she will grow more accustomed to that.
There were some good suggestions about dealing with childhood fears at KidsHealth: Anxiety, Fears, and Phobias. I especially liked that they mentioned "don't cater to fears", as in if the child is afraid of dogs don't cross the street to avoid them. That only reinforces the idea that there is something to be afraid of. Safe, supported exposure to the subject of your fears is something that normally works for everyone.
Now, that having been said, there are some cases where safe exposure just isn't enough. If you've tried "normal" methods for dealing with childhood fears and it either isn't working or your child's fears are getting worse, it may be time to get some help from a professional. Especially if you have no idea where your child's fears came form (you didn't mention anything about that).
We were pretty sure we knew or could guess at the source of our son's fears. Lack of familiarity was the root cause of most of them. In a Chinese orphanage nobody sleeps alone (sometimes 3 to a bed) and animals are food, not pets. Even puppies and kittens. And they didn't have many doors in their building.