My understanding was that they appeared with regular density

Regular density? Depending on what you mean by "regular" .. There is the famous prime number theorem. If π(n) is the number of primes less than n, then π(n) tends to n/ln(n) as n grows to infinity. Thus you could say that the density if primes within {1,..,n} is ln(n)/n. But this density gets smaller and smaller as n gets larger, thus the primes have to be getting (on average) farther and farther apart.

Also, as blazar mentions below. If you want to find a number X such that the first prime after X is at least X+200, just let X be 201! (that's with a factorial). Now X+1 may be prime, but X+2 is not prime since 2 divides X and 2. X+3 is not prime since 3 divides X and 3, etc... all the way until X+201.