I used normal paper from my notebook to perform filtration. The filtration is 100% perfect, but takes incredibly long to happen, because the paper is too dense.
I'm not planning to buy filtration papers. They're cheap, but I don't filter something every day, so they'd be laying around useless. I would prefer to be able to use some household item as a filtration paper.

One idea I had was to apply some substance on normal writing paper to make it softer.

I have tissues which are very soft, but these are too soft and will leave paper chunks in my filtrate.

Do you have any idea what should I use for filter paper, or how could I alter normal paper to suit my needs?

A practical alternative to separate such a slurry from the liquid phase would be to substitute filter and funnel by a centrifuge. With some luck you still find some with a manual crank (with conic tubes, about $$\pu{20 mL}$$ each). Based on own experience, this often is faster than setting up and then performing a suction filtration over a glass filter. On small scale, it is simpler to wash the slurry, and clean up the tools, too. Just keep the opposite tubes balanced and don't mind the surprised gaze of your colleagues around.