Alex P has given some great options, but I think this point is worth emphasizing: Play cards that are individually strong.
Quoting from the "Thoughtseize You" article linked by Alex P:
My philosophy in deckbuilding is to focus on cards that are individually powerful by their own merits. However, this is not the only way to build Magic decks. In fact, my friend and teammate, the esteemed Sam Black, takes much the opposite approach, looking for advantages in favorable interactions between cards. Such an approach has many benefits. Even my Tarmogoyfs will look foolish when they face down Sam’s Bonescythe Sliver, Megantic Sliver, Syphon Sliver, and Galerider Sliver all in play at once! (I insult Sam’s creativity only in the interest of a simple example).
It’s cards like Thoughtseize that allow the well-rounded, Rock-style strategies to compete against the (typically) more powerful synergy-based decks. Thoughtseize represents a way to trade resources quickly and efficiently, and once you can force that Galerider Sliver to stand on its own, your more powerful cards will win their individual battles. The Thoughtseize effect accomplishes the goal of breaking up synergy better than more situational answers like Doom Blade or Naturalize because of its applicability against creature-based strategies, control, and combo alike.
The point here is that Tarmogoyf is an individually powerful card - it's often a 2-mana 4/5 or better - but if the opponent is able to assemble all of Bonescythe Sliver, Megantic Sliver, Syphon Sliver and Galerider Sliver against it at the same time, then Tarmogoyf is just not good enough. Discard is a defense against these synergistic decks because if you take out one of these Slivers, suddenly all the other Slivers gets weaker, possibly to the point where Tarmogoyf is once again the strongest creature on the board.
Extrapolating from this you should find that when your deck is dependent on synergy, targeted discard is at its best against you. In fact, discard is very good against Griselbrand Reanimator. If you Thoughtseize them and see a hand of land, Dark Ritual, Griselbrand, Mind Twist and Lotus Petal, you can take the Mind Twist and suddenly the rest of their hand does nothing. For the same reason, you should find that your opponent's discard strategy is stronger against your Elves deck than your Red deck.*
But if that's the case, why is your opponent beating you anyway? The reason is simply that your red deck is playing underpowered cards. A 2-mana 2/2 haste is not Legacy viable! In a format as powerful as Legacy a 2 damage/turn clock is not fast enough; your opponent can easily take some damage before eventually reanimating Griselbrand, a card that single-handedly beats you. In the same way, cards like Trumpet Blast, Rakdos Cackler, Hellrider - they are just too slow for Legacy.
If you want to play monored in Legacy, take a look at Legacy Burn. It's not a top-tier deck, but it should be very advantaged against anyone trying to play discard spells. After all, they might make you discard one burn spell, but chances are you're going to draw another one next turn anyway.
*NB: by the way, your opponent likely misbuilt his deck. The point is that Mind Twist does not help him reanimate Griselbrand. If he spends all his fast mana on Mind Twisting you, then the Griselbrands are going to be dead in his hand. He is likely better off focusing on the primary gameplan of animating Griselbrand, and if he has to adopt a plan B, he's likely better off using the fast mana on a big creature that threatens to win the game on its own. Last I saw, the standard creature used in this way is Grave Titan.