Do the middle letters have to be next to each other?

No. They only have to appear in the given order somewhere between the first and last letter, with any number of other letters around them.

Words starting with L and ending in H

There are 69 of them. The shortest listed are labyrinth, larch, lash, latch, laugh.

69 words fit this spine. They're all listed below. They start with L and end with H, with anything in between.

A near miss

lakh, lash, lath, lech, lich, loch, loth, lush

laich, laigh, laith, larch, latch, laugh, leach, leash, leech, letch, loach, loath, lotah, lough, lunch, lurch, lymph, lynch

lamedh, latish, launch, lavish, length, liroth, loofah, lowish

ladyish, largish, larkish, lazyish, leftish, longish, loudish, loutish, lumpish

ladyfish, languish, lemonish, lightish, lionfish, littlish, liverish, logomach, lumpfish, lungfish

labyrinth, laccolith, lagomorph, lickerish, locksmith, logograph, logogriph, loincloth, loudmouth

largemouth, lithograph, lockstitch, luftmensch

lamellibranch, leprechaunish, liebfraumilch