Well time-dilation and length contraction are two sides of the same coin - one man's time-dilation is another man's length contraction! All the experiments that confirm time-dilation also confirm length contraction.
Consider a cosmic ray hitting our atmosphere, creating a muon with a half life that means it shouldn't be able to reach the ground in the time available. We measure a lot of these muons hitting the ground, because of their relativistic speed. From our point of view, the muon is time-dilated - time slows down for the muon, from our point of view, allowing it to exist for a longer time than it would exist if it were "at rest". But from the muons frame of reference, time is flowing normally, and it is the atmosphere of the Earth that is length contracted in the muons direction of travel. Either way, the muon reaches the ground when it's half-life says it doesn't have enough time to reach the ground.
As for the increase in mass, I refer you to Taylor and Wheeler:
"The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."
Ad relativistic mass - equations derived from it, namely relativistic energy and momentum, exactly match the energies we observe during particle collisions in accelerators.
For example, according to Newton, a proton travelling at the speed of light (asymptotic case) would have an energy of just 470 MeV. Yet we have accelerated protons to over 7 TeV, an energy 4 orders of magnitude higher. Since we know for a fact that they did not cross the speed of light the only other variable that's there to take the blame is their mass. And there we go.