Logo in IE11 Looks Very Low Quality

Hey santanin,
I have taking a look and IE11 doesn’t render images well because it’s using an old engine, the solution is suppose to be putting X-UA-Compatible IE=Edge in the header, so I put this code to the end of your child theme functions.php file in Appearance > Editor:

add_action( 'send_headers', 'add_header_xua' );
function add_header_xua() {
	header( 'X-UA-Compatible: IE=edge,chrome=1' );
}

and cleared your site cache, but didn’t notice a difference.
I then uploaded a smaller logo and used a media query to target IE11 only in your child theme css:

@media all and (-ms-high-contrast: none), (-ms-high-contrast: active) {
span.logo a:before {
content: '';
background: url(https://yoursite.com/wp-content/uploads/2018/01/tclogoie2.jpg) no-repeat center;
display: block;
height: 78px; 
}
}

While the image looked good in IE11 my attempts to hide the old logo failed, IE11 is limited as it is not css3 ready.
Overall IE11 is hard to work with and why Edge is it’s replacement, your logo looks good in Edge.
I would suggest trying some other sizes for your logo, I know you don’t want to go down to 340px, but 1400px is way to big. Perhaps try 700px, it should still look good on other browsers and may work with IE11. If you try a few sizes I’m sure you will find a good compromise.

Best regards,
Mike