I read this book in my school's library in 2017, so I'm guessing it was published some time around then.
I think it was about a young boy who got sold to a wizard, or his dad left him in the woods then a wizard found him and he was taken to a very cluttered cottage. The boy turns out to be very good at sorcery, I think, and he ends up going to a city with a large magical tower.
The cover was also the silhouette of a male standing in front of a building, and there were vines around the edges.
Jinx by Sage Blackwood?
In the Urwald, you don’t step off the path. Trolls, werewolves, and butter-churn riding witches lurk amid the clawing branches, eager to swoop up the unwary. Jinx has always feared leaving the path—then he meets the wizard Simon Magus.
Jinx knows that wizards are evil. But Simon’s kitchen is cozy, and he seems cranky rather than wicked. Staying with him appears to be Jinx’s safest, and perhaps only, option. As Jinx’s curiosity about magic grows, he learns to listen to the trees as closely as he does to Simon’s unusual visitors. The more Jinx discovers, the more determined he becomes to explore beyond the security of well-trod paths. But in the Urwald, a little healthy fear is never out of place, for magic—and magicians—can be as dangerous as the forest, and soon Jinx must decide which is the greater threat.
It doesn't exactly fit your description but it's close. Jinx gets lost in the forest, the Urwald, when he goes into the forest with his stepfather Bergthold. His stepfather is taken by trolls but Jinx is rescued by the wizard Simon and taken back to Simon's cottage.
The cover is Jinx standing in front of a building with vines round the edge, though it's not a silhouette. The date fits as well since the book was published in 2013.
Jinx doesn't go to a city, but he does go the tower of an evil wizard called the Bonemaster and that may be the tower you remember. Simon's cottage is described as:
They were sitting at the kitchen table on top of the big stone stove, which filled half the kitchen and was just pleasantly warm underfoot. Onions, dried apples, and pumpkin hung from the rafters overhead. There were cats everywhere, lying on barrels and shelves—there was one curled around the water pump.
so it is pretty cluttered though I don't think the word cluttered is actually used.
This sounds like a snippet from Magician by Raymond E Fiest.
The young boy is Pug, who is an orphan.
The opening chapter has him getting lost in the woods and being found by Kulgan ( The local magician) and taken back to his cottage to wait out the storm. The cottage is cluttered with magical objects and Pug demonstrates an affinity for the objects which causes Kulgan to take him on as an apprentice.
Much (much) later in the story Pug ends up as a trainee at the Assembly, a collective of Magicians on another world. The final stage of his testing involves standing on the top of a spire on the top of a tower in the city of Magicians.
The most common cover doesn't feature vines, but does have a nature like feel, with the silhouette of a man in the centre.
