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By reading Wikipedia page on braid group one finds the Larence-Krammer representation. This makes one wonder whether there are maybe simpler faithfull representation for small $n$ and indeed, reading the corresponding paragraph, it seems that there is hope given by Burau representation which was conjectured to be faithful for a long time. However, it is not faithful for $n\geq 5.$ This last Wikipedia page links to a review article by Faithful representations of the braid groups by Vladimir Turaev in which he states that the Burau representation is indeed faithful for $n\leq 3$ and that the case $n=4$ is open. So was there any progress in the last almost 20 years since the article was written? Quick googling suggests that some people are still working on it and that they seem to believe that it is indeed faithful. See the work of Beridze and Traczyk and Attacks on the Faithfulness of the Burau Representation of the Braid Group $B_4$ by Abdulrahim and Chreif.