Your puppy needs you to be there for him
Been bringing my 4 month old westie to the dog park, which contains a "small breed" section. I've been having a hard time getting him to play with other dogs - he either stays near me or goes off to chew on leaves. Eventually the other dogs stop chasing him since he doesn't reciprocate any interest in them and also outruns them easily (I make him run continuously at least 10 min a day on top of his 1 hr walk). I keep positioning myself so that he has to pass through other dogs to get to me.
I take him to the big dog section as well so that he gets maximum "initial interest" time from as many dogs as possible. He almost outruns even the big dogs, but squeaks when they catch up and push him around. When he runs to me, I jump onto a high rock that he claws at while trying to evade the other dogs, tumbling in the dirt in frantic pivots. Is there any risk of injury from him playing with the big dogs? So far all the other dogs are friendly but I'm worried that one "play" bite from an aggressive big dog will decapitate my baby...
I think you're trying to do what's best, but NONE of this is good for your puppy.
First off at 4 months old he's still very young and if he gets a bad experience he could be fearful his entire life of other dogs...or whatever it is that's giving him bad experiences. You should make sure 100% that your puppy is happy and feels completely safe with you and that you'll protect him. If you're jumping on a rock when he runs to you for help you're literally letting him down and letting him know that you're not there for him when he needs you. To gain confidence he needs to explore things that he is comfortable with AND he needs to know you're there to "save" him and get him out of trouble by picking him up or removing him from the situation. You're letting him fend for himself and simply put, that is not what you should be doing.
It sounds as if he's afraid, and that makes sense. He knows you're not going to get him out of bad situations. By hiding behind you and jumping up on the rock he's begging you to help him and keep him safe. Dogs that are afraid seek shelter under tables and benches, or behind people. Those things SHOULD be available to him. If he's overwhelmed, pick him up and "save" him. You need to be there for him every time. Read up on dog body language. There are videos on youtube about dog body language and some of them are at dog parks. They show dogs being chased around and trying to seek shelter and sometimes their person does the right thing and breaks up the conflict and helps the dog that's in distress, but other times the dog is on it's own and you can easily see that the dog is not happy and is not having fun. They never will have fun if they're always afraid.
Your little guy can very easily get hurt with these adult dogs. Even the smaller ones. Smaller dogs are often more likely to bite than big dogs. I think the recommended age for going to a dog park is six months.
About "making him run continuously every day"...Puppies are not supposed to put stress on their joints. There are a lot of articles online about puppies and the dangers of too much exercise. You should read up on this so you're fully informed.
"Generally, you shouldn't start running with your puppy before six months of age; any earlier and you can risk affecting his growing joints and muscles. Some large and giant breeds may not be ready until later."
At the dog park you "position myself so he has to go through other dogs to get to me"...again, it's so important that he learns that you'll protect him. Don't hide, don't avoid, don't jump on rocks. Be there to snatch him up when he's stressed. Be there to let him hide behind your legs when he's overwhelmed. If he never has an "out" from feeling overwhelmed then he'll NEVER be a confident dog. He'll be afraid, or he'll learn he has to defend himself and he'll possibly get agressive. If he's not happy going to the dog park, then hold off from taking him. Take him places that are safer and more comfortable for him, like the puppy play time get "classes" that Petco has.
You said he yelps when the big dogs are around him. That IS a sign of stress. Get him out of that situation.
You asked if there is risk of injury. YES. About a month there was a big fight at a dog park here. Both dogs ended up injured badly. One lost an eye. I've heard that a dog was killed at a dog park. Some people bring dogs to the dog park that they're not sure of. Dogs that are a risk to all of the other dogs. My sister's dog was hurt and forever traumatized from being attacked when he was younger. You need to be alert and watch learn the body language of other dogs. If you see one acting up and causing any problems it's not a situation to take chances with.
Keep in mind that your puppy is a baby. Don't throw him to the wolves. If you want a happy, confident dog it starts with letting him know that you've got his back.