If there is a path that goes across each bridge once and only once, then each land mass needs an even number of bridges connected to it, as the bridges must come in pairs(one to get in, one to get out). The exceptions are the land masses you start and end on, as they can have an odd number of bridges. This is because you only have to get off the land mass(in the case of your starting point) or get on the mass(in the case of your ending point). Except for those two, all other land masses must have an even number of bridges connected to them. Since all the land masses in the problem have odd numbers of bridges connecting them, it is impossible to make such a route which crosses each bridge once and only once.

As Akshansh said, there is no way to walk across each bridge only once. For any system of this type, to be able to traverse each bridge only once, you need to have either two land masses with an odd number of bridges connected to them, or no land masses with an odd number of bridges connected to them. Otherwise, you would have to traverse at least one bridge twice. It is impossible to have a system in which only one land mass has an odd number of bridges connected to it. If one of them has an odd number of bridges connected to it, then there must be a second with an odd number of bridges connected to it. This problem, called the 7 Bridges of Koenigsberg, was originally proved to be unsolvable by Euler in 1735, and his proof sparked the subjects of graph theory and topology. Here is a link to a page from the Mathematical Association of America discussing Euler’s proof.
https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem