@MiloBrandt already showed a working strategy and a proof, but I thought some of the arguments can be simplified a bit. Since I didn't want to suggest an edit to his legitimate solution, decided to add my version below.


**As Milo already pointed out, a good strategy for the policeman would be to encompass the 4 blocks of the town in a clock-wise manner one by one, clock-wise.**

**Since the policeman is twice as fast as the thief, if the thief is in the center of the town at some point, then there exists a moment in which the policeman is in the center, and the thief is not on the boundary, i.e. he gets shot.**

**Now, assuming the thief never visits the center, his angle with respect to the coordinate system defined by the two middle roads changes continuously. The angle of the policeman with respect to the same coordinate system can be defined to change continuously as well. Since the policeman needs less time to increase his angle with 360 degrees than the thief, there will be a moment when the two have the same angle. However, this implies that the policeman will be able to see the thief, and shoot him.**

P.S. If the moderators or Milo decide it is better to edit the original solution using this one, feel free to do so and erase this answer.