Normalising floating-point numbers will most likely be done in hardware. It will use bit operations, but not visible to the programmer using it. And normalising will be done in very close conjunction with rounding.
Floating-point numbers usually consist of a sign bit, an exponent field, and a mantissa field. Since floating-point numbers are supposed to be normalised, which means the highest bit of the mantissa is supposed to be 1, this is often not stored. In addition, you will have tiny numbers that are denormalised (they are so small that if they were normalised, you wouldn't be able to store the exponent), and you have special values like infinities and not-a-numbers.
In floating-point arithmetic, usually the first step is detecting special values and handling them. For example, +inf / NaN = NaN, +inf * any positive number = +inf, etc. etc. There are lots of special cases, but you just follow the definition what the results should be. For multiplication and division, zeroes may also be handled as a special case.
For multiplication and division, you normalise denormalised numbers. You count the leading zero bits in the mantissa, then shift the mantissa to the left to have the leading bit set, and subtract the shift count from the exponent. You add or subtract the exponents and perform the multiplication division, giving a larger mantissa than needed. Depending on the inputs, the highest bit of the result mantissa may be cleared. In that case you shift the result mantissa to the left and decrease the exponent. Now you perform rounding based on the extra mantissa bits you have. Rounding can lead to overflow, in which case the exponent is increased. Finally you check the exponent: If it is too large, the result becomes infinity, If it is too small, the result will be a denormalised number, possibly zero.
For addition and subtraction, you need to have identical exponents. So you shift the mantissa with the smaller exponent to the right. Then you perform the addition or subtraction. You may have a huge number of leading zero bits, so you shift the mantissa to the left by possibly many bit positions and change the exponent. You don't change it to something lower than the exponent for denormalised numbers. And then you perform rounding, again checking for overflows.
So the bit operations that you perform are: Extracting sign bit, exponent and mantissa, processing sign bits. Normalising denormalised numbers by counting leading bits, shifting, and adjusting exponents. Normalising results again by counting leading bits etc. Rounding is done using logical operations.