How can I suppress bad breath after eating garlic or onion?
Raw veggies are your friend, the more chlorophyll (the green stuff) the better. I say raw because the fibers clean your teeth. The chlorophyll reacts with the sulfur and neutralizes the bad smell. Just keep chewing and you'll be fine.
Eating vegetables if of course not a replacement for proper mouth hygiene, but interestingly with garlic it is more effective to chew parsley than to brush your teeth.
Nothing can really deal with allium breath. The sulfur compounds in the alliums are absorbed into the bloodstream via the gastrointestinal tract, then released into the alveoli of the lungs, where they are exhaled along with carbon dioxide waste when you breathe out. The smell isn't coming from your mouth, it's coming from your lungs. Some of it may also appear in your sweat. None of these recommendations will work. The only way to avoid the smell is to cook the onion/garlic until the sulfurous compounds break down.
I think it makes a lot of difference how you cook with the garlic (or onion). If you put the fresh garlic on a very hot flame for a minute or two, and then put the fire down, you keep all the good attributes of the garlic, but dramatically reduce the smell it creates from your mouth.
Bad breath could also come from other problems. Getting advice from a dentist can help there.
You should drink milk. It is much better than trying to rinse with water. It is actually proven, the study was published in the journal of food science.
The most important sentences fo the abstract are:
Fat-free and whole milk significantly reduced the head-, mouth-, and nose-space concentrations of all volatiles. Water was the major component in milk responsible for the deodorization of volatiles.
Milk was more effective than water and 10% sodium caseinate in the deodorization of allyl methyl sulfide, a persistent garlic odor, in the mouth after garlic ingestion.
The work being a bit theoretical, they also insist that mixing the garlich with milk before consuming it results in less bad breath than if you drink milk after eating garlic. In fact, I often don't have the possibility or inclination to do it, and I would be wary to extend the findings to other dairy products (if they had tried yogurt insead of milk, we'd know that tzatziki is a more business-friendy form of garlic than a tomato sauce).
The whole study is available online, no paywall.