No, that's not the case at all.

If you fall into a black hole, you can do no experiment to detect the horizon, it's a completely unremarkable region of space to you. Infinite gravity is only really a thing at the singularity, but that's almost definitely just because our theoretical models breakdown and stop giving accurate descriptions of reality there.

The stretching is just because of tidal forces, which means that gravity gets so much stronger closer to the black hole that your feet are pulled harder than your head, you experience the same thing standing on earth, it's just that the change in gravity is basically negligible here.

Source: Was a black hole physicist for a while

Small edit: Tidal forces stretch you in the exact same way that they stretch the ocean, thus creating ocean tides.