The magnetic field is the Lorentz transform of the electric field. If we boost to the reference frame of the moving charges in your current, an electric field will emerge. The resulting electric force is perpendicular to both the magnetic field and the velocity of the charges in the current. Let's see an example:
There is a test charge $q$ moving with velocity $v$ to the right in the lab reference frame. A short distance below it is a linear charge distribution with linear charge density $\lambda$. Superimposed on this is another linear charge distribution with linear charge $-\lambda$, which is moving with velocity $u$ to the right. Because $\lambda-\lambda=0$, there appears to be no electric field in the lab frame. Now boost to velocity $v$ to the right so that the test charge $q$ is at rest. The positive line charge experiences relativistic length contraction such that the distance between charges on the wire shorter and the charge density is higher. The new positive charge density is now $\gamma \lambda$, where $\gamma$ is the relativistic time dialation / length contraction factor. The negative line charge, however, was already moving to the right, and was already length contracted. By boosting, we undo some of the length contraction and therefore reduce the linear charge density of the negative group of charges. Now the density of positive charges does not equal the density of negative charges anymore and the net charge is no longer zero! An electric field emerges! The electric force on the test charge, which is stationary in this reference frame, is $F=qE$, and this force is perpendicular to the boost velocity (which is the test charge velocity in the lab frame) and to the magnetic field that we saw in the lab frame.
There are still magnetic fields in the boosted frame, but they are different and a piece has been converted into an electric field.