The mode where the obvious move is wrong
Bridge gives you two words and asks for the shortest path between them. COLD to WARM. There is no clock. The header tells you the par before you start.
It is the quietest mode in the game and the one people get most annoyed by, and both of those come from the same place.
What you naturally do, and why it fails
Handed COLD and WARM, almost everybody does the same thing. They look at the target, find a letter it has that they do not, and go and get it.
COLD has a W missing. So: CORD, WORD. Now there is a W. Then WARD, and WARM. Four morphs, and that happens to be par, and it feels like the method worked.
Try it on a pair where it does not. The instinct is to reduce the difference between the word in front of you and the word you want, one letter at a time, and to treat any move that increases the difference as a wasted turn. That instinct is right most of the time, which is exactly what makes it dangerous, because the pairs where it fails do not announce themselves.
The trouble is that the two words are not neighbours in the dictionary just because they are neighbours on the page. To get from here to there you have to pass through real words, and the real words available around you have nothing to do with where you are going. Sometimes the only road out runs the other way for a while.
Three kinds of pair
Every Bridge you are given is one of three characters, and the game names which after you finish rather than before.
Gentle ones let you walk more or less straight there.
Standard ones get you to the target and cost you a morph or two on the way. You will reach it, you will not get gold, and afterwards you will see the shorter route and it will be obvious.
Devious is the one the mode is built for. Somewhere in it there is a move that looks wrong and is the only way through, and the road that looks right runs out under you. The first move on a devious pair frequently looks like a mistake: it changes a letter the target does not care about, or it makes the word longer when the target is shorter.
The reason the character is only named at the end is that saying it up front would give the puzzle away. Telling you "the obvious move is wrong here" is most of the answer.
How to actually approach one
Start by not touching anything. There is no clock. Every second you spend looking is free, and Bridge is the one mode where the first move is the most expensive one to get wrong.
Work backwards from the target. This is the single most useful habit and hardly anybody does it. Ask what a word could look like one move before WARM. WARD, WART, WORM, WARY. Now you have four destinations instead of one, and some of them are much easier to reach than WARM is.
Count the letters. If the two words are different lengths, at least one of your morphs has to be an add or a remove, and you know that before you start. If they are the same length, you might still need one of each, and a pair that seems to need six replacements when par is four is telling you a shortcut runs through a shorter word.
Check which positions actually differ. COLD and WARM differ in all four. HEAD and TAIL differ in three. The count of differing positions is the absolute floor on how many morphs you need, so if par is higher than that count, the extra morphs are going somewhere and the puzzle is about finding out where.
Say the awkward move out loud. If you have been stuck for a minute, take the move you dismissed first and play it. On a devious pair the way through often starts with something that looks like giving ground.
The grade
Reaching the target in par is gold. One over is silver. Two over is bronze. Reach it any longer than that and you have crossed, which counts, and takes no multiplier.
Par is the true shortest path and not an estimate. If it says four, four is possible, and the game checked before it posed the puzzle. That matters when you are three moves in and convinced the thing is impossible: it is not, and the number on the header is not being generous with you.
Gold roughly doubles what the run pays, so the grade is most of the score. This is the mode where thinking longer is straightforwardly worth more, and there is no speed multiplier and no streak to lose by pausing.
The crossing
Once a day there is a different kind of Bridge. A named pair, chosen for what it means rather than for how far apart the words are, and sometimes dedicated to a real bridge somewhere in the world. It arrives with its name and one line about how the two words are related, and if there is a story behind the dedication you are told it once you reach the far bank and not before.
Those can run longer than an ordinary Bridge, up to ten morphs. They are held to the same standards as every other pair, which means a beautiful pair that fails them does not ship, and there are a number of lovely ones that did not.
This is the oldest idea in the game, older than the game. Lewis Carroll's word ladders were jokes about their endpoints: PIG to STY, APE to MAN, FLOUR to BREAD. The pleasure was in the pair, and the ladder was how you earned the punchline. A daily crossing is that, with a bridge in it.
Why it is worth the annoyance
Blitz asks whether you can find a word. Bridge asks whether you can find a route, which is a different question and a slower one.
The moment the mode exists for is narrow and quite specific. You are stuck, you have tried the obvious thing twice, and then you see that the way through runs backwards. Nothing else in the game does that, and it is worth being annoyed for.