There are two main strategies you can use here:
Negative Feedback Loops
These are "rubber banding" mechanics that give players who are lagging an advantage to help them catch up, or slow the progress for players who are already ahead. This reduces the impact of an early lead, and helps keep players in the running right up to the end. Items in Mario Kart are a good example of this.
In a quiz game this could include things like:
getting a wrong answer deducts a percentage of the player's score (impacting high-score players more aggressively)
giving trailing players a bonus question to answer, or a bonus guess if nobody gets the right answer in a round, or a follow-up question based on the previous (more scoring opportunities for players who need a hand)
if nobody answers a question, one of the players currently in the lead is forced to answer (so an early lead is a liability for penalties)
a "steal" mechanic that lets a player with a right answer choose who to steal some points from (players will disproportionately pick the leader). You Don't Know Jack's "screw" mechanic that lets players force an opponent to answer functions similarly.
Escalating Stakes
The other major strategy is to make later rounds more valuable than early rounds. So while an early lead is good, players who are left behind aren't out of the running - a few wins late in the game can turn the tables, even against an opponent with a strong lead.
You'll often see this done in the form of questions worth progressively more points (see Jeopardy with its player-controlled progression down the columns n the board), a "second half" round of questions that score double, or a final question / minigame like YDKJ's Jack Attack with much higher scoring potential than any previous round.
