imc

We can see that a $1$-step staircase requires $4$ toothpicks and a $2$-step staircase requires $10$ toothpicks. Thus, to go from a $1$-step to $2$-step staircase, $6$ additional toothpicks are needed and to go from a $2$-step to $3$-step staircase, $8$ additional toothpicks are needed. Applying this pattern, to go from a $3$-step to $4$-step staircase, $10$ additional toothpicks are needed and to go from a $4$-step to $5$-step staircase, $12$ additional toothpicks are needed. Our answer is $10+12=\boxed{22}$