NowThese days, turtles are threatened by nets used in shrimp hunting.

"These days" or "Nowadays" is more natural than "Now"

In the lecture, the professor said that a turtle excluder device, which is called a TED, is useful to save endangered sea turtles.

You might want to give additional context to the lecture - e.g. "In a lecture I recently attended..."

Firstly, shrimp boats seldom catch turtles and it is rare, so they do not havlike to use TEDs because using TEDs makit causes some loss of shrimps and it costs additionallyncreases costs.

The professor argues that there are thousands of shrimpers in the US, so ifalthough it is rare to trap a turtle when it comes toyou consider just one boat, it becomn total the problem creates enormous damage to endangered sea turtle species.

The wayapproach is to shorten the time limit to keep the netsfor which the nets are kept under the sea.

You could also join this with the previous sentence, e.g. "Next, there is another method to save turtles which is more reasonable than TEDs: keeping the nets under the sea for a shorter period"

However, the professor said that thatis restriction is impossible because the sea police cannot see every shrimp boats.

On the other hand, forcing tohe use of TEDs is more controllable and easier than the ideawould be easier to implement.

I think this is a simpler way of making the same point!

The professor insists that it iswould be easy to modify and develop a larger size of TEDs and these, which would enable larger turtles to survive.

If the larger TEDs don't yet exist, it's better to say "would be easy" (if they do exist, you can use "is easy")

Now, tTurtles are threatened by nets used in shrimp hunting.

Beginning a text with 'Now' comes across as informal. It is more a feature of spoken English than of formal written English.

In the lecture, the professor said that a turtle excluder device, which is called a TED, is useful to save endangered sea turtles.

I would add the indefinite article (a/an) before TED.

Firstly, shrimp boats seldom catch turtles and it is rare, so they do not have to use TEDs because using TEDs makcauses some loss of shrimps and it cshrimps to be losts additionallynd it increases costs.

'Seldom' and 'rare' mean the same thing, so it is unnecessary to use both at the same time.

The professor argues that there are thousands of shrimpers in the US, so if it iswhile it may be rare to trap a turtle when it comes to one boat, it becomes enormously damageing to endangered sea turtle species when all shrimpers are taken into account.

Theis way is to shorten the time limit to keep the netsduring which nets may be kept under the sea.

However, the professor said that thatis restriction is impossible because the sea police cannot see every shrimp boats.

The repetition of 'that that' is not necessarily wrong, but it is usually preferable to avoid it. In this case, it makes sense if you change the second 'that' to 'this'.

A noun following 'every' is always singular.

On the other handWhereas/By contrast, forcing tohe use of TEDs is more controllable and easier than the ideaalternative.

Use 'on the one hand/on the other hand' to compare the advantages of two options, the disadvantages of two options, or the advantages and disadvantages of one option.

"On the one hand, this is an advantage of option A; on the other hand, this is an advantage of option B."

"On the one hand, this is a disadvantage of option A; on the other hand, this is a disadvantage of option B."

"On the one hand, this is an advantage of option A; on the other hand, this is a disadvantage of option A."

In this case, you have use it to contrast a disadvantage of one option with an advantage of another.