The phrase "successive weeks" means that something happens continuously or one after another in each week, without any breaks or gaps in between. Full definition
Five artists will present their work over successive weeks and the show will evolve out of those presentations. (artguide.pro)
Ashley Barnes scored the equaliser, his second goal in successive weeks at Turf Moor, to set us on our way to a 2 - 1 win against Everton last Saturday and that, along with his overall performance, has won him both the man of the match vote and the player ratings. (uptheclarets.com)
Luis Suarez would usually have received the MOTM award for his two expertly taken goals, but I am going to give the award to Jose Enrique for a second successive week. (eplindex.com)