glitich in the outputs

there is nothing wrong with 1 process style - it can actually help prevent errors. Plus it would register all your outputs for you. I think older books only recommend the 2 process style because at the time compilers were pretty crap at doing anything with source code outside a few templates. Now they are much much better.

Library ieee;
use ieee.std_logic_1164.all;
PORT (Right,Left,Hazard : IN STD_LOGIC;
  Clk, Reset : IN STD_LOGIC;
  LA,LB,LC : OUT STD_LOGIC;
  RA,RB,RC : OUT STD_LOGIC);
END Vehicle_Tail_Lights_control;
ARCHITECTURE state_machine OF Vehicle_Tail_Lights_control IS
  TYPE state_type IS (Idle,L1,L2,L3,R1,R2,R3,LR3);
  attribute ENUM_ENCODING: STRING;
  attribute ENUM_ENCODING of state_type: type is
  "000 001 011 010 101 111 110 100";
  SIGNAL current_state : state_type;
BEGIN
 
  PROCESS(clk, reset)
  BEGIN
    if reset = '1' then
      current_state <= idle;
      
    elsif rising_edge(clk) then
      
      CASE current_state IS
        WHEN idle =>
          Lc<='0';Lb<='0';La<='0';Ra<='0';Rb<='0';Rc<='0';
          IF ( (Hazard='1') OR ((Left='1') AND (Right='1')))
          THEN current_state <= LR3;
          ELSIF ( (Hazard='0') AND (Left='0') AND (Right='1'))
          THEN current_state <= R1;
          ELSIF ((Hazard='0') AND (Left='1') AND (Right='0'))
          THEN current_state <= L1;
          ELSE
            current_state <= Idle;
          END IF;
            
        WHEN L1 =>
          Lc<='0';Lb<='0';La<='1';Ra<='0';Rb<='0';Rc<='0';
          IF (Hazard = '1') THEN current_state <= LR3;
          ELSE current_state <= L2;
          END IF;
            
        WHEN L2 =>
          Lc<='0';Lb<='1';La<='1';Ra<='0';Rb<='0';Rc<='0';
          IF (Hazard = '1') THEN current_state <= LR3;
          ELSE current_state <= L3;
          END IF;
            
        WHEN L3 =>
          Lc<='1';Lb<='1';La<='1';Ra<='0';Rb<='0';Rc<='0';
          current_state <= Idle;
          
        WHEN R1 =>
          Lc<='0';Lb<='0';La<='0';Ra<='1';Rb<='0';Rc<='0';
          IF (Hazard = '1') THEN current_state <= LR3;
          ELSE current_state <= R2;
          END IF;
            
        WHEN R2 =>
          Lc<='0';Lb<='0';La<='0';Ra<='1';Rb<='1';Rc<='0';
          IF (Hazard = '1') THEN current_state <= LR3;
          ELSE current_state <= R3;
          END IF;
            
        WHEN R3 =>
          Lc<='0';Lb<='0';La<='0';Ra<='1';Rb<='1';Rc<='1';
          current_state <= Idle;
          
        WHEN LR3 =>
          Lc<='1';Lb<='1';La<='1';Ra<='1';Rb<='1';Rc<='1';
          current_state <= idle;
          
      END CASE;
    end if; --clk
  END PROCESS;
END state_machine;