A commonly quoted figure is that 1-2 pounds (0.5-1 kg) a week is a healthy rate at which to lose weight, and that 'crash diets' are bad for you. What is this figure based on, and in what way is losing weight faster than this unhealthy?
The quoted "1-2" pounds per week (or approximately 0.5 - 1 kilo) is based on the traditional 3500 calories burned = 1 pound of fat. Now, while the exact figure of 3500 calories is somewhat open to debate, that is the rationale. 500 calories a day = 1 pound of loss per week.
The problem with crash dieting is twofold. One, the majority of the weight lost is water weight, and will rapidly return once the person hydrates themselves. The second problem is that in an effort to lose large amounts of weight in a short time is that people will essentially starve themselves, and possibly do so while adding in exercise.
In the very short term, this may not pose any problems for a healthy person, but if the dieter is compromised in any way health wise, or the practice is continued for a longer period then nutrient deficiencies can start arising, which can introduce a long litany of health problems.
Another problem is that people that lose large amounts of weight in a short time tend to binge eat once the diet period is over, and regain most if not all the weight. Many even end up weighing more than they did before they did the extreme dieting.
There have been studies comparing many of the popular diets such as Atkins, South Beach, China Study, Paleo, and on and on. They have all shown that weight loss is a simple factor of calorie reduction. Eat less calories than you expend, and you will lose weight. Period. (And "starvation mode" is a myth.) When you first start dieting, you may lose more than the 1-2 pounds a week, which is normal. After a bit, your body will readjust and the weight loss will slow down, however if you maintain the calorie deficit, you will eventually lose the weight.
Annals of Internal Medicine (Short abstract)
The Lancet (Not a study, but references two studies)
A systematic review of low-carbohydrate diets found that the weight loss achieved is associated with the duration of the diet and restriction of energy intake, but not with restriction of carbohydrates. Two groups have reported longer-term randomised studies that compared instruction in the low-carbohydrate diet with a low-fat calorie-reduced diet in obese patients (N Engl J Med 2003; 348: 2082–90; Ann Intern Med 2004; 140: 778–85). Both trials showed better weight loss on the low-carbohydrate diet after 6 months, but no difference after 12 months.
I want to add an answer focusing on the second part of your question
in what way is losing weight faster than this unhealthy
There's a few unhealthy things that can happen to people who lose a lot of weight quickly. It's of course subjective what rapid is. The most severe effects can be seen in people on very low calorie diets (VLCD).
Gallstones develop significantly more often in people losing weight rapidly, often requiring surgery. Thus is linked to not getting enough fat from food.
In an early study, gallstones developed in 25% of patients during 8 weeks of VLCD, and 6% of patients eventually required cholecystectomy (19). In a second trial, asymptomatic gallstones occurred in ∼12% of patients within 6 months of starting a VLCD, and approximately one-half of these individuals eventually became symptomatic, requiring cholecystectomy
Another risk is "simple" malnutrition. Not getting enough food leads to not getting enough nutrients.
There can be cardiac complications, hairloss, dizziness, constipation, and muscle cramps.
Very low calorie diets are of course far from the normal way people diet. But they show what can happen.