Ask your vet about Revolution as this (in the proper dosage level) has been used in puppies. I’m surprised the vet didn’t mention it.
I’m also assuming that you are speaking about fleas as hundreds of ticks and no fleas is highly unlikely.
The reason you keep seeing more of them is because new ones are constantly hatching as there are several stages to their life cycle.
If you can treat your yard where the pup goes that will help long term. I know that there is some type of product to treat the yard with that is not dangerous to the pup. It contains nematodes which feed on flea larvae.
For the immediately, this is what worked for me. Get a roll of masking tape and an extra fine flea comb in a color other than black or brown. Petco has some really cheap plastic ones that have small teeth on one side and REALLY small narrow spaced teeth on the other side
Assuming you’re right handed, tear off several pieces of masking tape about 2–3 inches long. Hold one piece in your left hand and get ready to immediately slap it onto the comb’s teeth after you run the comb thru the pups coat.
This sticks to the fleas before they have a chance to hop away. Fold in on itself trapping them.
Take another piece of masking tape and keep going. You get into kind of a rhythm after awhile. It worked for me and afterwards I had the satisfaction of squishing and popping them to death in their sticky tape graves. There was just something viscerally satisfying about that.
It’s more involved to try to explain it to someone over the internet than it is to just do it. :)
I did this with two wiggling cats for several months and it saved me the trouble and mess of trying to deal with both the animals and a bowl full of dish detergent toppling over.
Hopefully your vet has also checked the pup for other parasites and the possibility of anemia? If not consider looking for another vet
Also ask him about Revolution. And just remember that this is the height of flea and tick season. It can only get better from here.
Also vaccuum daily and put mothballs in the bag to kill the critters otherwise you are just giving them a cozy breeding and hatching place. Change bags more frequently even if not full. Good luck.