First shot fired is superimposed by second shot or simply disappears after second shot and etc

I am trying to resolve the cause of my triggered bullet disappearing if in a short period of seconds I fire a second bullet. I imagine that screen.fill(GRAY) might be a cause and it should be inside the Class Bullet() or maybe I need to use all_sprites_list.update() elsewhere.

Here is my MWE code:

import pygame
BLUE = (0, 0, 255)
GREEN = (0, 255, 0)
GRAY = (128,128,128)
SCREEN_WIDTH = 800
SCREEN_HEIGHT = 600
class Cowboy(pygame.sprite.Sprite):
      def __init__(self):
          super().__init__()

          self.image = pygame.Surface([14,20])
          self.image.fill(BLUE)
          self.rect = self.image.get_rect()
class Bullet(pygame.sprite.Sprite):
      def __init__(self):
          super().__init__()
          self.image = pygame.image.load("bullet.png").convert()
          self.rect = self.image.get_rect()
      def update(self):
         self.rect.y += -3
def main():
    pygame.init()
    size = [SCREEN_WIDTH, SCREEN_HEIGHT]
    screen = pygame.display.set_mode(size, pygame.RESIZABLE) 
    all_sprites_list = pygame.sprite.Group()
    bullet_list = pygame.sprite.Group() 
    enemies_list = pygame.sprite.Group()
    bullet = Bullet()
    enemy = Cowboy()
    enemies_list.add(enemy)
    all_sprites_list.add(enemy)
    enemy.rect.y = 50
    enemy.rect.x = 50
    done = False
    clock = pygame.time.Clock()
    while not done:
        for event in pygame.event.get():
            if event.type == pygame.QUIT:
                done = True
            elif event.type == pygame.MOUSEBUTTONDOWN:
                bullet_list.add(bullet)
                all_sprites_list.add(bullet)
                bullet.rect.x = 340
                bullet.rect.y = 200
        screen.fill(GRAY)
        all_sprites_list.draw(screen)
        clock.tick(60)
        pygame.display.flip()
    pygame.quit()
if __name__ == "__main__":
    main()