The fifth gym leader is actually in an old town that you had visited previously. The fifth gym leader is Norman in Petalburg city. In order to beat him I would bring a good Pokemon with good fighting moves, otherwise you wouldn't survive against Slaking or Vigoroth.
Winona at fortree city.
The fifth Gym is in Petalburg against your father.
in the gym
beat him with your pokemon!
saffron city (psychic type gym leader)
simply go back to petalburg where your father is gym leader and you must challenge him
Gym 5 is located in Safaron City, and the Gym Leader is Sabrina. She uses Psychic type Pokemon, and it's a bit of a puzzel to get to her.
cianwood city.
you us a rod at the waters edge
1: the first gym is rock 2: the 2nd is fighting 3: the 3rd is electric 4: the 4th gym is fire 5: the 5th is normal 6th: the 6th is flying 7th: the 7th is psychic 8: the 8th is water
roxanne, brawly, wattson, flannery, norman.
The Relic Badge is given to you by Gym Leader Fantina of Hearthome City (Gym #5).