I've been vegetarian for many years, and tried veganism for a while too. There is no doubt about it, if you follow healthful vegetarianism it is worlds better for you than the standard american diet.
BUT...and this is a huge BUT...vegetarianism can be JUST AS UNHEALTHY as a regular diet. Many veggies live off tons of grains and mock-meat type foods, so while their saturated fat intake may be low, their blood sugar levels and insulin resistance may still be crappy. Domino's cheese pizza is vegetarian. Ice cream is vegetarian. Candies are often vegetarian. Chocolate soymilk has just as much sugar in it as regular chocolate milk. If you go plant-based, it should be fruit/veggie based, not processed grain based. Your protein sources should be the real deal - tofu, beans, lentils, etc - not vegetarian hot dogs, chicken nuggets, and fake meats. If you find yourself eating pasta/grain heavy dinners 3+ times a week, you might have a problem with balance in the diet. Most of the studies showing benefits of vegetarianism follow vegetarian cultures/groups who traditionally eat a very whole foods based diet anyway, not the processed vegetarianism diets often found now in middle class Western industrialized nations.
Personally, I do not believe that vegetarianism or veganism are that much better for you than if you eat a whole foods diet that happens to include meat and fish. As someone who has explored many types of dietary lifestyles, the biggest health issue seems to be over consumption of sugar and processed foods, not one particular type of food group.
It is also important to understand that while vegetarianism has many benefits, much of the science is exaggerated or espoused by those who also follow the diet for very strict moral reasons. So scrutinize what you read - vegetarian groups often twist or exaggerate findings, or withhold information on other diets that include meat/fish, because it is in their best interests to do so. I am a huge fan of animal welfare, but the information pushed is often more propaganda than nuanced scientific discussions. Perfect example is the fact that the groups never talk about the sheer scale of mass killing caused by the soy industry, even though it is a well documented fact that soy production costs billions of animal lives every year. A block of tofu shipped from California is probably more environmentally damaging and morally dubious than the pastured pork raised and slaughtered by your neighbor down the road. But pro-veg groups can't discuss this because so many vegetarians rely heavily on soy, and because some members might take to eating locally grown meat every now and then if they think it is less damaging than buying a block of tofu. Both would be detrimental to their message and recruitment efforts, so they just don't discuss it. The same thing happens with scientific findings surrounding saturated fat and meat consumption.
Basically, if you are getting your science from a group that's entire mission and reason for existing is vegetarianism, then you might not be getting the true story. Just as you wouldn't trust the Cattleman's Association for an objective analysis of the benefits and risks of eating beef, don't trust a pro-vegetarian group to give you the objective truth about the benefits and risks of vegetarianism.