I had a basenji whose aggression toward almost any dog on earth, even those she grew up with, were legendary. Her favorite target was my chow/coyote, who thankfully never tried to hurt her-- just get her off. She finished her championship with a 5 pt major with stitches on her head. The judge said to the handler "nice stitches." Her only friend was my male rottweiler. She semi tolerated her niece. I have never loved a dog as much in my life, but she was horribly animal aggressive. So after her championship, I spayed her. So yes, there are dog aggression issues not uncommon at all in the breed.
That said, the fact that your dog was hurt may be the root of the over the top to the point of even going after puppies (typically should not be seen as a threat.. so I don't agree with Pat. Not liking, snarking at pups.. sure. AGGRESSIVELY going after them... no). And for any type of aggression being on a leash makes it worse, but you have no choice. What you need to do for his and other animals' safety is get him used to walking with a basket muzzle. As far as needing exercise running free, you do know that many dogs live long happy lives without being off leash? Teaching to run with a bike or other activity, finding a friend who has a yard they'll let you access to play fetch or use a lure toy etc... there are ways to exercise without being loose around other animals.
However, your dog's aggression is a sign of severe stress and he can't be happy. What I suggest is you consider a behaviorist to work, not with making him dog-friendly... but with learning to ignore other dogs. You can do some of the work on your own if you are dedicated and willing to slowly work on it. The process is much like any work on teaching a dog to "leave it" and de-escalate. The steps are very simple for desensitization.
First, work at home on leave it command. A lot. Until you have it 100 percent. Mary's lessons are great for general training. The top is your concern right now, but exercise is both mental and physical. So the 2nd link is one way to do the mental ... links on right and left.
Start working on him with a basket muzzle and walking. It is critical you can get him calm and that the basket muzzle allows you to give him a small treat and that he can drink in it. Once he is calm with it around home and dog free walks, you're ready to begin desensitizing.
At least 3 times a week, get help with people with dogs.. you can go to a park where dogs are NOT running loose and have a friend help you. Tell people to not let their dogs come within even about 20 feet of your dog, but to walk by without looking and throw a small treat. Do a few a day. Come back and keep at that distance until he stops going ballistic. Move about 4 feet closer (you can even mark the distance). Repeat process. You should be getting pretty quick response to "leave it" by now. Use leave it, treat yourself as you see a dog approaching. Have them move closer... repeat. Keep this up until they are able to get within about 6 feet without reacting. At this point, you want one honking great kind of treat... steak, chicken, apple (my dogs think its better than steak!), whatever his most prized food is.. but keep it still fairly small. You want the dogs to move closer, but you now will be the source of all treats using "leave it" so he doesn't look a them. By this point, you should have far less reaction.
Now, it's your turn to walk. Engage the buddy system, but with YOU walking your dog past sitting so standing but not moving, people with dogs. This should go much faster. Once you can get within 5 feet without response, you move to both walking past each other about 10 feet out, then move closer.
The goal of course is not going to make him like other dogs, is not going to make him okay off lead around them, but getting him to at least tolerate will make his and your life better.
IF his only issue is dog reaction, then at this point you are ready to talk to obedience teachers about taking him to a class. Even though you should by now have mastered all the basics, his socialization by working around other dogs in a confined setting is great. If he has other issues, then really consider a behavior therapist, maybe even prozac. If the dogs are it, then it's simply making the next several months your life working toward him unlearning his response is what is needed.