Tavern Brawler feat states that:
You are proficient with improvised weapons and unarmed strikes.
And the rules for improvised weapons are:
In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus. An object that bears no resemblance to a weapon deals 1d4 damage.
So can you use Magic Weapon spell on an Improvised weapon?
Best Answer
This depends on DM ruling.
The rules don't explicitly cover this case, so there are two, equally-valid ways to rule this.
The first is that improvised weapons don't count as weapons. This ruling is supported mainly by a few parts of the Improvised Weapons description (page 147, PHB). It says:
These line heavily imply that, while improvised weapons often act like weapons, they actually aren't. Since magic weapon specifies that it can only target weapons, by this ruling improvised weapons can't be made magic.
This interpretation is supported by Jeremy Crawford, who tweets:
The second interpretation is that, since the phrase "improvised weapons" contains the word "weapon", then improvised weapons must be weapons. With this interpretation, anything that can be done to a weapon can also be done with an improvised weapon.
Both of these interpretations are valid, and based on valid rules reasoning, so this is going to be dependent on your DM.