We study self-avoiding walks on restricted square lattices, more precisely on the lattice strips $\mathbb{Z} \times \{-1,0,1\}$ and $\mathbb{Z}\times \{-1,0,1,2\}$. We obtain the value of the connective constant for the $\mathbb{Z} \times \{-1,0,1\}$ lattice in a new shorter way and deduce close bounds for the connective constant for the $\mathbb{Z}\times \{-1,0,1,2\}$ lattice. Moreover, for both lattice strips we find close lower and upper bounds for the number of SAWs of length $n$ by using the connective constant.