There are several ways to solve the problem of objects that are larger than your tiles.
Perhaps the easiest way is just to add the entity to every tile that it intersects with. Essentially the entity is in a wide range of tiles at any time. The is simple to implement but doesn't scale well for performance if your entities vary greatly in size.
A more flexible solution would be to use hierarchical tiles, where you have several tile sizes, and you place entities into the set of tiles that best fits their size. This works well if you have a fixed number of different entity sizes.
The most flexible solution would be a quadtree, which starts with very large tiles and recursively splits them so that they always fit the smallest object.
I drew a diagram of some simple cases to illustrate the ideas. The green boxes are the tiles that the gray object would be added to in that tile structure.