Vector help: Multiple bullets acting as one

I'm trying to make a 2d shooter with SDL and I got as far as having multiple bullets but they act as one when a bullet goes off screen. I shoot 1st bullet, then I shoot 2nd bullet. 1st bullet goes off screen (deletes) and 2nd bullet disappears. I shoot 3rd bullet, 2nd bullet reappears where it disappeared and repeat.


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class Bullet
{
public:
	static const int BULLET_WIDTH = 15;
	static const int BULLET_HEIGHT = 25;
	static const int BULLET_VEL = 5;

	Bullet();

	//xFire, yFire - position relative to player
	void handle_input(int xFire, int yFire);

	void bullet_render(); //renders bullet
	void bullet_move(); //moves bullet

	bool isAlive;

private:

	//bullet offsets
	int bullet_x, bullet_y;
	
	//velocity of bullets
	int velX, velY;
};
std::vector<Bullet> bullets;

Bullet::Bullet()
{
	isAlive = false;

	bullet_x = 0;
	bullet_y = 0;

	velX = 0;
	velY = BULLET_VEL;
}

void Bullet::handle_input(int xFire, int yFire)
{
	if(event.type == SDL_KEYDOWN && event.key.keysym.sym == SDLK_z)
	{
		Bullet b;
		bullet_x = xFire;
		bullet_y = yFire;
		isAlive = true;
		bullets.push_back(b);
	}
}

void Bullet::bullet_move()
{
	if(isAlive == false) return; //do nothing

	for(int i = 0; i < bullets.size(); i++)
	{
		bullets[i].bullet_x += velX;
		bullets[i].bullet_y -= velY;
	}

	//if bullet goes off screen, delete
	for(int i = 0; i < bullets.size(); i++)
	{
		bullet[i].bullet_y -= velY;
		if((bullets[i].bullet_y < 0) || (bullets[i].bullet_y + BULLET_HEIGHT > SCREEN_HEIGHT))
		{
			isAlive = false;
			bullets.erase(bullets.begin() + i);
		}
	}

}

void Bullet::bullet_render()
{
	if(isAlive == true)
	{
		for(int i = 0; i < bullets.size(); i++)
		{		
			apply_surface(bullets[i].bullet_x, bullets[i].bullet_y, bullet_fire, screen);
		}
	}
}

int main(int argc, char* args[])
{
	bool quit = false;

	//offsets of bg
	int bgX = 0;
	int bgY = 0;

	Player player(SCREEN_WIDTH/2 - (55/2), SCREEN_HEIGHT - 65);
	Player player2(50, 50);

	Bullet zbullet;

	Timer fps;

	//initialize
	if(init() == false)
	{
		return 1;
	}


	//load the files
	if(load_files() == false)
	{
		return 1;
	}

	while(quit == false)
	{
		fps.start();

		while(SDL_PollEvent(&event))
		{
			player.handle_input();
			zbullet.handle_input(player.x+20, player.y);

			if(event.type == SDL_QUIT)
			{
				quit = true;
			}
		}
		
		//scrolls background
		bgY += 2;

		if(bgY >= background->h)
		{
			bgY = 0;
		}

		//renders text
		message = TTF_RenderText_Solid(font, "SCORE", textColor);

		if(message == NULL)
		{
			return 1;
		}
		
		//shows bg and text
		apply_surface(bgX, bgY, background, screen);
		apply_surface(bgX, bgY - background->h, background, screen);
		apply_surface(10, 0, message, screen);
		
		//moves objects
		player.move(player2.get_rects());
		zbullet.bullet_move();
		
		//shows objects
		zbullet.bullet_render();
		player.render();
		player2.render();
		
	if(SDL_Flip(screen) == -1)
	{
		return 1;
	}


	//Cap the frame rate
	if( fps.get_ticks() < 1000 / FRAMES_PER_SECOND )
	{
	SDL_Delay( ( 1000 / FRAMES_PER_SECOND ) - fps.get_ticks() );
	}
	}

	clean_up();

	return 0;
}


I'm guessing the problem is in the bullet_move function, but I just don't know what to do. Can anyone help me understand to fix this?
Try not erase items in a vector in 'for' loop.
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