how to beat random pawn push.
In early or mid game, generally, I say let them. Focus on piece activity instead, away from the pawn storm. Delay dealing with the storm as long as possible, even letting them initiate pawn trades first so you get that one more tempo before dealing with it. Pawn storms are powerful, but very tempo expensive. Just vigilantly watch for tactics as always
In the end game it's a much bigger deal. Kings are great against pawn storms. Bishops as well. You don't necessarily have to solve the pawn storm long term, but you should increase the amount of tempo they'll have to spend to get it working so that you have tempo to do your own plans before anything becomes a queen.
These answers are pretty abstract but feel free to post an example of a pawn push you lost against for more tangible advice.
Random pawn push is quite risky. Every pawn push makes its weaknesses because pawns cannot go back again. Every time your opponent push his pawns, those pawns are moving away from his territory - which means it becomes harder for your opponent to support his pawns.
The reason why pawn push feels difficult to deal with is it makes you to have less space.(which often leads to lack of space for piece development)
The best way to deal with pawn push is to defend with your pawn. The best piece to deal with pawn is a pawn. Push your pawn to limit the movement of your opponent's pawns. When you think your opponent seems to be pushing his pawns too much, you can make a less pawn move and develop your pieces first. This is one option that let your opponent take the center, focus on your development first and then fight back later. (This is also the idea for hyper modern openings. King's Indian is one example.)
Don't forget that every time your opponent pushes his pawns, he loses opportunity to develop his piece and create weaknesses in his position.