The transformation of sand into sandstone per se cannot be directly witnessed, as it's happening deep in the earth. However, there are types of cementation that happen quickly and in plain view:
A ScienceDirect article on Cementation says:
Cementation is the precipitation of a binding material around grains, thereby filling the pores of a sediment. Berner (1971, p. 97) states that “cementation by silica must be predominantly a phenomenon of later diagenesis because almost no examples are found in recent marine sediments.” In contrast, cementation by calcium carbonate occur rapidly after deposition. A good example is beachrock, a mix of beach and intertidal sand (usually carbonate and skeletal fragments) cemented by CaCO3 in subtropical to tropical climates. The cement forms so rapidly that human artifacts only a few decades old are commonly found cemented into the beachrock.
-- David R. Montgomery, ... Henri Spaltenstein, International Geophysics, 2000 quoted at sciencedirect.com
I have witnessed an early stage of this personally. When I was a kid, my brother and I used to risk typhoid by visiting a stream a couple of blocks from our house. I'd walk out on a sandbar left behind by a recent flood and pick up a fist-sized quartz rock that looked collectible.
But the underside of the rock sometimes had smaller pebbles and sand grains cemented to the underside. They couldn't be brushed off-some mineral had connected the grains to the larger rock. There were clumps of cemented sand grains and pebbles in the sandbar.
This mineral might have been clay minerals derived from the feldspar that used to hold the quartz together, but in this urban environment, calcite leached from concrete and iron oxide were also possible.