Can a level 3 fighter use their shield as an improvised weapon and attack using their bonus action on top of their regular attack?
Yes, you can use your Shield as an improvised weapon
The damage would be 1d4 plus your Strength bonus, the attack your Strength bonus, without your proficiency bonus.
No, you can not use it in addition
As it is not a light weapon, you can not attack with any other weapon in that turn unless you have the Dual Wielder feat.
What you could do
If you take Dual Wielder and Tavern Brawler, you can attack with a shield using a bonus action, and add your proficiency bonus to the attack roll.
What you should do
Take the Shield Master feat, get better DPR (advantage on attacks, plus Dueling Style) and better control, for half the price (spending only one feat).
Wearing the shield - No
If you are wearing the shield (you've spent 1 round donning it and you are getting the AC bonus) then you cannot use it as an improvised weapon, as it is now worn, not wielded.
In this case a shield bash is the same as a booted kick or a gauntleted punch or a helmed headbutt - it is an unarmed attack.
If a GM does rule that a worn shield is an improvised weapon, doing 1d4, then they also have to rule that so is a suit of armour, clothing, gloves, boots, and so on. It is simpler to rule that "worn items are armour, not weapons" and "carried items are weapons, not armour". The distinction being that carried items can be picked up and dropped using a free action but worn items require actions to don and doff.
Further evidence for armour and weapons being separate: if a character wears armour that they are not proficient in then they suffer penalties (Basic Rules, page 44) but if a character holds a weapon they are not proficient in then they suffer no penalties.
Carrying the shield - yes
If you've just picked the shield up and are using it to hit someone then you can use it as an improvised weapon (it would be similar to a wagon wheel, which is mentioned in the Basic Rules). You will attack without proficiency (unless you have the Tavern Brawler feat). It does 1d4 + STR bludgeoning damage.
In both cases, the shield is not a weapon, so two-weapon fighting does not apply.
I would rule that the second case requires two hands, so two-weapon fighting would definitely not apply.
To clarify, my reading is that an improvised weapon does not count as a "weapon" for features that say things like "When you attack with a weapon…". The errata for unarmed attacks has clarified that though unarmed attacks involve making a melee weapon attack, they are not weapons because unarmed does not appear on the (errata) weapon table.
If you do rule that improvised weapons in general are "weapons" then you have to determine for each weapon whether or not it is light, heavy, two-handed, thrown, etc. This is so you can answer questions like "does great weapon fighting apply to hitting someone with a wagon wheel?" and "does two-weapon fighting apply when I have a chair in one hand and a large book in the other?".
Note that if you rule that a shield is a weapon, then dueling fighting style does not allow the use of a shield, contradicting Crawford.