Where is the anvil in Lumbridge?
Old School RS is the 2007 version of Runescape (in fact, the tag states that). According to the Lumbridge page on the Wiki:
Since the update on 24 June 2010, there is an anvil located near the furnace, which is convenient, since players no longer have to run to the Draynor Village sewers or Varrock to train Smithing or make armour/items.
The version of the game you are playing does not have this anvil there since it does not contain this update. It has everything up until August 10, 2007, with the addition of some other updates that originate from polls that Jagex runs and the players elect to have in the game. The God Wars Dungeon was also added into Old School RS, as it was not present as of August 10, 2007 (it was released a few weeks later), and a few other miscellaneous things as well.
Edit: As of July 9th, 2020, a rusted anvil exists in Lumbridge, right next to the furnace. However, it can only be used to smith bronze bars.
There is no anvil in Lumbridge. You can easily find some anvils in Varrock. There is also a bank just a few steps away, and it's low-level-friendly.