Of 10 consecutive numbers, exactly five are even, one or two are multiples of six and three or four ar emultiples of three. Thus at least one of the multiples of three is odd. This leaves at most four numbers that are neither divisible by two nor three. Therefore, unless $2$ and $3$ appear themselves among the primes, no five primes can occur.
For the problem with four primes, there are seemingly many solutions, e.g. if we require additionally that the first of the ten numbers is prime, I find 3, 5, 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, 5651, 9431, 13001, 15641, 15731, 16061, 18041, 18911, 19421, 21011, 22271, 25301, 31721, 34841, 43781, 51341, 55331, 62981, 67211, 69491, 72221, 77261, 79691, 81041, 82721, 88811, 97841, 99131, 101111, 109841, 116531, 119291, 122201, 135461, 144161, 157271, 165701, 166841, 171161, 187631, 194861, 195731, 201491, 201821, 217361, 225341, 240041, 243701, 247601, 247991, 257861, 260411, 266681, 268811, 276041, 284741, 285281, 294311, 295871, 299471, 300491, 301991, 326141, 334421, 340931, 346391, 347981, 354251, 358901, 361211, 375251, 388691, 389561, 392261, 394811, 397541, 397751, 402131, 402761, 412031, 419051, 420851, 427241, 442571, 444341, 452531, 463451, 465161, 467471, 470081, 477011, 490571, 495611 when looking only up to 500000.