It’s a good question. One answer is the lack of High Explosive (HE) shells. In 1914, shrapnel shells were far easier to produce because the metallurgical and chemical components were more straightforward. Indeed, within the British army, the first HE shell for the 18-pounder field gun (the standard field artillery piece for the British and Dominion forces during the First World War) was only tested in October of 1914, two months after the beginning of the war. In all countries, the manufacture of artillery shells couldn’t keep pace with the rate of expenditure and even as late as the end of 1915, there was a backlog of thousands of HE shell casings within British munitions factories because the procedure for filling them was so complex. British production of HE shells only caught up with the production of Shrapnel shells in March of 1916. The lack of HE shells made destroying earthworks and trenches difficult.

What happened after the manufacturing of HE shells caught up with expenditure? Not a lot. Obviously the destructive capabilities of artillery increased dramatically but by 1916 both sides were firmly entrenched. We can use the example of the Somme to demonstrate the failure of artillery as the sole means of destroying defences. The preceding artillery bombardment was intended to destroy German defences and cut the barbed wire. It did neither. The German’s garrisoning the front line survived the bombardment in their deep and well-constructed dugouts and emerged (though shaken), ready to defend their line which they did to great effect to the detriment of the attacking troops. Techniques such as the Moving Barrage or Creeping Barrage had been developed but not perfected and the barrage frequently moved faster than the infantry could to keep up with it, leaving them exposed. Greater cooperation between artillery units and the infantry was needed but due to poor communication, this wasn’t possible until later in the war.

The other thing that needs mentioning is that there were relatively few troops in the front lines except when an attack was looming. The majority of troops were held back as reserves to be moved up if the enemy attacked but until then were relatively safe from artillery fire. If the front line was hit, those troops would retreat to the safety of their dugouts as the Germans did on the Somme and emerge when the barrage had lifted.

Later in the war, artillery would play a far more tactical role, shifting from a role purely bent on the destruction of enemy trenches and fortifications to a role of neutralisation. By 1917 the creeping barrage had largely been perfected and had become a regular tactic during attacks against trench lines. As the artillery barrage crossed trench lines, the infantry following close behind would enter and clear the trench while the artillery prevented enemy reinforcements from reaching their beleaguered comrades. While effective, the tactic didn’t always translate into success. The German defensive lines were deep and attacking units were inevitably worn down clearing trenches and were often unable to exploit and breakthrough they may have occurred. This changed in 1918 when the Hindenburg Line was breached. A combination of artillery, tanks and infantry attacked German lines. While one infantry unit cleared and held a trench line, another fresh unit would continue forward, continuing the pressure put on German defenders. This was known as leapfrogging.

Hopefully that answers your question. Like all things, practice makes perfect and the use of artillery in the type of warfare seen on the Western Front went through a huge learning curve. Unfortunately while we usually view failure as a learning opportunity, failure in this case meant the deaths of thousands of men.

Source: Battle Tactics of the Western Front: The British Army’s Art of Attack, 1916-1918 by Paddy Griffith.

The Great War by Les Carlyon

Edit: Corrected year

There are two main reasons that I can think of as to why artillery didn't render trench warfare useless.

Early in the war an issue that the British in particular faced was the reliability of shell fuses. Leading up to an assault like the beginning of the Battle of the Somme, artillery was easily capable of firing tens of thousands of shells over a few days. However, only a portion of those shells would actually explode when they hit the ground. Typically this was caused by the mud being too soft and the shell would just bury itself without triggering the fuse. The issue was sensitivity. There is anecdotal accounts that after the massive pre-Somme bombardment it was possible to walk across no-man's-land on a carpet of unexploded shells.

The second reason why artillery didn't render trench warfare useless ties into the sensitivity of fuses and reliability of shells. If you fire tens of thousands of shells at the enemy leading up to an assault, the shells will churn up the ground to an unusable level. This became an issue during the now infamous Battle of Passchendaele, when the already damaged drainage systems in the area were shelled and saturated the constantly agitated soil.

These two factors worked together to degrade artillery's status from a presumed war winner to just one part of the combined arms tactics necessary to succeed on the Western Front.

You seldom find a single weapon platform completely dominate warfare.

Yes, artillery was important, but you're still firing an artillery shell. Not only had chemistry not entirely caught up with the new weapon system (the Incendiary Round and similar high-explosive weapon ordinance weren't quite there- TNT had only been realized to be a very useful explosive ordinance about a decade prior) but you were using a very inaccurate weapon platform to hit a very narrow target- slit trenches made it very difficult to hit what you wanted.

Furthermore between the dead zone artillery and machine guns induced, on the off chance saturated artillery strikes cleared an area, there'd be enough time for new troops to flood in, set up, and repel any charge.

The real issue was that the internal combustion engine hadn't yet come into it's own. Tanks (and lighter vehicles) were grossly ineffective at their job, and aircraft still couldn't manage a noteworthy load.

Fusing had a lot to do with it. The VT fuse wasn't invented until WWII and even then it was considered so valuable that it wasn't allowed to be used if there was a possibility of it falling into enemy hands (until the battle of the bulge). The VT fuse uses a primitive form of radar to provide a consistent air burst at a constant height above ground. This is incredibly effective because the explosion of a shell is predominantly spherical so when it explodes in the air it can do damage to a lot of targets on the ground, including targets inside a trench or foxhole. Without VT fusing a shell either has a timer to explode in the air or explodes on contact. Timers are extremely difficult to get right and rely on a combination of many shells to eventually get a lucky hit that does the damage you want. Ground burst shells have the problem that a lot of their explosive power is uselessly directed upwards or downward. They dig a crater and provide a small radius of lethal damage. But heavily dug in targets that are just a few meters away might survive injury.

To take out a target in a trench you would need a direct hit, and then only the soldiers in that small section of the trench would be killed (since trenches were designed with zigzags to protect against artillery hits).

Artillery is tremendous for taking out exposed targets. If you have, say, a building, an airfield, a warehouse, a train station, etc. then artillery can smash it in a matter of hours. But trenches are actually one of the best defenses against artillery, aside from completely underground tunnels.

Two additional things ( that are not mentioned by the other comments):

1. With a trench you need to hit the trench in order to be effective. For illustration ( that is I am guessing the numbers) lets say that a artillery shell will kill ( or badly wound) anybody within 5m of the explosion. In open field this are roughly 80 m^2, in a 1m broad trench this are just 10 m^2.

2. The trench systems were not simple trenches, but had bunkers incorporated into the trench, so during a artillery barrage the opposing soldiers would not actually be in the trench, but in the bunker. ( And only take position in the trench when the barrage stopped.